diff --git a/ChangeLog.md b/ChangeLog.md
--- a/ChangeLog.md
+++ b/ChangeLog.md
@@ -1,5 +1,59 @@
# Changelog for lazyppl
+## 1.1.0
+
+### Added
+
+- `lazyppl-examples`: separate examples and plotting package
+- Playground: interactive browser models
+- Example pages: locally served fonts and icons
+
+### Changed
+
+- Build: Cabal and GHC 9.6.7, 9.12.4, 9.14.1
+- `gp`, `wiener`: pure Haskell without LAPACK
+- `mh`, `mhirreducible`: lower stream memory retention
+- `memrec`: `Ord` keys without `Show`
+- Integer `memoize`: restricted to `Prob`
+- Distributions: validate inputs and parameter domains
+- `beta`, `dirichlet`: reject unsupported shape parameters
+- Random labels and acceptance draws: open interval (0,1)
+- `mh`, `mhirreducible`, `weightedsamples`, `wis`: separate generator per run
+- `superuniformbounded`: bounded graph angle sampling
+
+### Fixed
+
+- `wis`: resample from all weighted samples
+- Integer `memoize`: return values at their own indices
+- `mh`, `mhirreducible`, `wis`: zero and invalid weights
+- `mh1`: reject proposals without reverse sites
+- `beta`, `dirichlet`: sampling at numerical boundaries
+- `beta`, `dirichlet`: reject unsupported custom tree labels
+- `categorical`: rounding with finite and infinite streams
+- `categorical`: accept finite probabilities with a rounding-sized shortfall
+- Dirichlet processes: small-stick accumulation
+- `poisson`: sampling at extreme labels
+- `ibp`: first customer's dish count
+- `gp`: initial and conditional distributions
+- `wiener`: variance at extreme times
+- `normalLogPdf`, `betaLogPdf`, `dirichletLogPdf`: extreme-input accuracy
+- `binomialLogPmf`: large-count accuracy
+- `gammaLogPdf`, `poissonLogPmf`: extreme-input accuracy
+- Memoization, Gaussian processes and counters: concurrent state
+- `uniformbounded`, `uniformBoundedLogPdf`: overflowing bound differences
+- Playground: retry failed WebAssembly downloads
+- Playground: validate results before plotting
+- Mondrian playground: reject overlaps and duplicate identifiers
+- GraphDemo: graph probabilities and figure edges
+- Example pages: correct maximum-likelihood sample labels
+- Example pages: syntax colours in dark mode
+- Website generation: preserve prior site on build failure
+
+### Removed
+
+- `LazyPPL.SingleSite`: moved to `lazyppl-singlesite`
+- `lazyppl`: `hmatrix`, LAPACK and plotting dependencies
+
## 1.0.1
- Add support for GHC 9.8 and 9.10
diff --git a/README.md b/README.md
new file mode 100644
--- /dev/null
+++ b/README.md
@@ -0,0 +1,74 @@
+# 
LazyPPL
+
+LazyPPL is a Haskell library for Bayesian probabilistic programming, drawing on the fact that lazy data structures are a natural idiom for programming with infinite-dimensional Bayesian methods such as Poisson/Gaussian/Dirichlet processes.
+
+The crucial semantic idea, inspired by developments in synthetic probability theory, is to work with two separate monads: an affine monad of probability, which supports laziness, and a commutative, non-affine monad of measures, which does not. This gives a Metropolis-Hastings implementation that works with lazy programs.
+
+The main LazyPPL modules are on [Hackage](https://hackage.haskell.org/package/lazyppl) and [Stackage](https://www.stackage.org/). Various worked examples in literate Haskell are at [https://lazyppl-team.github.io](https://lazyppl-team.github.io).
+
+Laziness lets us specify non-parametric models cleanly: processes or infinite-dimensional functions only have their finite explored portion materialised. We have first-class
+
+* stochastic memoization: `memoize :: (a -> Prob b) -> Prob (a -> b)`
+* Gaussian processes: `wiener :: Prob (Double -> Double)`
+* point processes: `poissonPP :: Double -> Double -> Prob [Double]` (in the example demos)
+* Dirichlet processes: `dp :: Double -> Prob a -> Prob (Prob a)`
+
+
+## Examples
+
+* [`WienerDemo`](https://lazyppl-team.github.io/WienerDemo.html) ([`lhs`](examples/models/WienerDemo.lhs)) is a simple implementation of regression using a Wiener process. Via maintaining a hidden table of previous calls, it appears to be a bona fide random function $R\to R$ constructed lazily. Some values are sampled during simulation, others during plotting.
+
+
+
+* [`RegressionDemo`](https://lazyppl-team.github.io/RegressionDemo.html) ([`lhs`](examples/models/RegressionDemo.lhs)) demonstrates piecewise linear regression. The change points are drawn from a lazy Poisson process.
+
+
+
+* [`Clustering`](https://lazyppl-team.github.io/ClusteringDemo.html) ([`lhs`](examples/models/ClusteringDemo.lhs)) contains some simple clustering examples, where the number of clusters is unknown. Stick-breaking is lazy, and we use stochastic memoization.
+
+
+
+* [`ProgramInduction`](https://lazyppl-team.github.io/ProgramInductionDemo.html) ([`lhs`](examples/models/ProgramInductionDemo.lhs)) is a simple example of program induction over an arithmetic language. Random expressions are represented as an infinite forest with a finite path through it.
+
+
+
+* Feature extraction via the [Indian Buffet Process](examples/models/AdditiveClustering.hs), and relation inference via the [Mondrian Process](examples/models/MondrianExample.hs) and a Chinese-Restaurant-Process-based [Infinite Relational Model](examples/models/IrmTest.hs).
+
+
+## Library
+
+* [`LazyPPL.hs`](src/LazyPPL.hs) provides the core monads and inference. `Prob` is an affine monad of probability measures. `Meas` is a non-affine monad of unnormalised measures with `sample` and `score`. The Metropolis-Hastings algorithm `mh` takes a probability `p` of changing any given site of the rose tree of random seeds. Setting `p=1/num_sites` approximates single-site MH and `p=1` recovers multi-site MH. We can't compute the number of active sites without forcing more of the lazy tree, so we use this `p`-parameterised form. A second algorithm, `mh1` in [`lazyppl-singlesite`](singlesite/), implements a single-site proposal kernel by inspecting the Haskell heap.
+* [`LazyPPL/Distributions.hs`](src/LazyPPL/Distributions.hs) and `LazyPPL/Distributions/*` provide common distributions and abstract types capturing exchangeability. For example, `newCustomer :: Restaurant -> Prob Table` for the Chinese Restaurant Process ([`DirichletP.hs`](src/LazyPPL/Distributions/DirichletP.hs)) and `Restaurant -> Prob [Dish]` for the Indian Buffet Process ([`IBP.hs`](src/LazyPPL/Distributions/IBP.hs)).
+
+
+## Installation
+
+Build with Cabal: [install GHCup](https://www.haskell.org/ghcup/install/) for GHC 9.14.1 and Cabal 3.18.2.0, then `cabal update` and `cabal build` in the repo root. First build takes a few minutes.
+
+Run a demo: `cabal run -fdemos wiener-exe` (or `regression-exe`, `clustering-exe`, etc.), or explore in the REPL with `cabal repl lib:lazyppl`.
+
+The library is pure Haskell and needs no LAPACK or BLAS installation. The plotting demos use the `matplotlib` Haskell wrapper, which calls into Python: `sudo apt-get install -y --no-install-recommends python3-matplotlib python3-numpy python3-scipy python-is-python3`.
+
+To try lazyppl in your browser without installing anything, click [](https://codespaces.new/lazyppl-team/lazyppl). This opens a ready-to-use VS Code environment with GHC and Cabal already in place.
+
+
+## Writing your own probabilistic models
+
+Under `executables:` in `examples/package.yaml`, add a stanza and run `hpack examples` (0.39.6) to regenerate `examples/lazyppl-examples.cabal`:
+
+```yaml
+executables:
+ # ...
+ mymodel-exe:
+ main: MyModel.hs
+ source-dirs: models
+ ghc-options:
+ - -threaded
+ - -rtsopts
+ - -with-rtsopts=-N
+ - -main-is MyModel
+ dependencies:
+ - lazyppl
+```
+
+Run with `cabal run mymodel-exe`, or open in the REPL: `cabal repl exe:mymodel-exe`.
diff --git a/lazyppl.cabal b/lazyppl.cabal
--- a/lazyppl.cabal
+++ b/lazyppl.cabal
@@ -1,15 +1,15 @@
cabal-version: 1.12
--- This file has been generated from package.yaml by hpack version 0.38.1.
+-- This file has been generated from package.yaml by hpack version 0.39.6.
--
-- see: https://github.com/sol/hpack
name: lazyppl
-version: 1.0.1
+version: 1.1.0
synopsis: Lazy Probabilistic Programming Library
description: LazyPPL is a Haskell library for Bayesian probabilistic programming. It supports lazy use of probability, which is useful for specifying non-parametric models, and we provide new Metropolis-Hastings algorithms to allow this. For illustrations, see .
.
- Note: the Gaussian process module (@LazyPPL.Distributions.GP@) uses @hmatrix@, which links against LAPACK. On Ubuntu install with @sudo apt-get install libgsl0-dev liblapack-dev@.
+ LazyPPL is pure Haskell and needs no BLAS or LAPACK installation.
category: Statistics
stability: experimental
homepage: https://lazyppl-team.github.io/
@@ -21,12 +21,17 @@
license-file: LICENSE
build-type: Simple
tested-with:
- GHC == 8.10.7
- , GHC == 9.6.7
- , GHC == 9.8.4
- , GHC == 9.10.1
+ GHC == 9.6.7
+ , GHC == 9.12.4
+ , GHC == 9.14.1
extra-source-files:
+ README.md
ChangeLog.md
+ test/reference/beta-laws.tsv
+ test/reference/beta-quantiles.tsv
+ test/reference/dirichlet-sticks.tsv
+ test/reference/log-densities.tsv
+ test/reference/reference.py
source-repository head
type: git
@@ -41,15 +46,187 @@
LazyPPL.Distributions.GP
LazyPPL.Distributions.IBP
LazyPPL.Distributions.Memoization
+ LazyPPL.Distributions.Mondrian
+ LazyPPL.Internal.History
+ LazyPPL.Internal.Sampler
+ LazyPPL.Distributions.GP.Pure
hs-source-dirs:
src
build-depends:
- base >=4.7 && <9
- , containers >=0.6.5.1
- , hmatrix >=0.20.2
- , log-domain >=0.13.2
- , math-functions >=0.3.4.2
- , mtl >=2.2.2
- , random >=1.2.0
- , transformers >=0.5.6.2
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite api-test
+ type: exitcode-stdio-1.0
+ main-is: Api.hs
+ other-modules:
+ StrictMemo
+ Api.Support
+ Api.Domains
+ Api.Distributions
+ Api.Processes
+ Api.Inference
+ hs-source-dirs:
+ test
+ build-depends:
+ QuickCheck >=2.19.0.0 && <2.20
+ , base >=4.18.3.0 && <4.23
+ , bytestring >=0.11.5.4 && <0.13
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , typed-process >=0.2.13.0 && <0.3
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite chain-test
+ type: exitcode-stdio-1.0
+ main-is: Chain.hs
+ hs-source-dirs:
+ test
+ ghc-options: -threaded -rtsopts
+ build-depends:
+ QuickCheck >=2.14.2 && <2.20
+ , base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , deepseq >=1.4.8.1 && <1.6
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , mtl >=2.3.1 && <2.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite concurrency-test
+ type: exitcode-stdio-1.0
+ main-is: Concurrency.hs
+ hs-source-dirs:
+ test
+ ghc-options: -threaded -rtsopts -with-rtsopts=-N2
+ build-depends:
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite gp-test
+ type: exitcode-stdio-1.0
+ main-is: GaussianCondition.hs
+ hs-source-dirs:
+ test
+ build-depends:
+ QuickCheck >=2.14.2 && <2.20
+ , base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite hidden-state-o2-test
+ type: exitcode-stdio-1.0
+ main-is: HiddenState.hs
+ other-modules:
+ KnownLimit
+ hs-source-dirs:
+ test
+ ghc-options: -O2 -threaded -rtsopts -with-rtsopts=-N2
+ build-depends:
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite hidden-state-test
+ type: exitcode-stdio-1.0
+ main-is: HiddenState.hs
+ other-modules:
+ KnownLimit
+ hs-source-dirs:
+ test
+ ghc-options: -threaded -rtsopts -with-rtsopts=-N2
+ build-depends:
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite logpdf-test
+ type: exitcode-stdio-1.0
+ main-is: TestLogPdf.hs
+ hs-source-dirs:
+ test
+ build-depends:
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite replay-test
+ type: exitcode-stdio-1.0
+ main-is: Replay.hs
+ hs-source-dirs:
+ test
+ ghc-options: -threaded -rtsopts -with-rtsopts=-N2
+ build-depends:
+ QuickCheck >=2.14.2 && <2.20
+ , base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
+ default-language: Haskell2010
+
+test-suite stream-test
+ type: exitcode-stdio-1.0
+ main-is: Stream.hs
+ hs-source-dirs:
+ test
+ ghc-options: -threaded -rtsopts -with-rtsopts=-M256M
+ build-depends:
+ base >=4.18.3.0 && <4.23
+ , containers >=0.6.7 && <0.9
+ , lazyppl
+ , log-domain >=0.13.2 && <0.14
+ , math-functions >=0.3.4.2 && <0.4
+ , random >=1.3.1 && <1.4
+ , transformers >=0.6.1.0 && <0.7
+ , vector >=0.13.2.0 && <0.14
default-language: Haskell2010
diff --git a/src/LazyPPL.hs b/src/LazyPPL.hs
--- a/src/LazyPPL.hs
+++ b/src/LazyPPL.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE GeneralizedNewtypeDeriving, ScopedTypeVariables #-}
+{-# LANGUAGE GeneralizedNewtypeDeriving, ScopedTypeVariables, CPP #-}
{-# OPTIONS_GHC -Wno-unrecognised-pragmas #-}
{-# HLINT ignore "Use second" #-}
@@ -23,7 +23,7 @@
c. 'wis' (simple reference weighted importance sampling)
- See also the SingleSite module for a separate single-site Metropolis-Hastings algorithm via GHC.Exts.Heap and System.IO.Unsafe.
+ See also "LazyPPL.SingleSite" in @lazyppl-singlesite@ for a separate single-site Metropolis-Hastings algorithm via "GHC.Exts.Heap" and "System.IO.Unsafe".
3. Various useful helpful functions.
@@ -35,7 +35,7 @@
Most of the structure here will not be needed in typical models. We expose more of the structure for more experimental uses.
-The `Distributions` module provides many useful distributions, and further non-parametric distributions are in `Distributions.DirichletP`, `Distributions.GP`, `Distr.IBP`, and `Distr.Memoization`.
+The "LazyPPL.Distributions" module provides many useful distributions, and further non-parametric distributions are in "LazyPPL.Distributions.DirichletP", "LazyPPL.Distributions.GP", "LazyPPL.Distributions.IBP", and "LazyPPL.Distributions.Memoization".
-}
@@ -43,7 +43,7 @@
module LazyPPL
( -- * Rose tree type
--
- -- | Our source of randomness will be an infinitely wide and deep lazy [rose tree](https://en.wikipedia.org/wiki/Rose_tree), regarded as initialized with uniform [0,1] choices for each label.
+ -- | Our source of randomness will be an infinitely wide and deep lazy [rose tree](https://en.wikipedia.org/wiki/Rose_tree), regarded as initialized with uniform choices from the open interval (0,1) for each label.
Tree(Tree),
-- * Monads
Prob(Prob), Meas(Meas),
@@ -63,7 +63,9 @@
import Data.Monoid
import System.Random hiding (uniform)
import Control.Monad
-import Control.Monad.State.Lazy (State, state , put, get, runState)
+import System.IO.Unsafe (unsafeInterleaveIO)
+import LazyPPL.Internal.History (History, newHistory, appendMutation, labelAt)
+import LazyPPL.Internal.Sampler (randomLabel, checkedLogWeight)
import Numeric.Log
@@ -103,7 +105,7 @@
newtype Meas a = Meas (WriterT (Product (Log Double)) Prob a)
deriving(Functor, Applicative, Monad)
-{- | A uniform sample is a building block for probability distributions.
+{- | A uniform sample from the open interval (0,1) is a building block for probability distributions. It is never 0 or 1, so an inverse transform such as @-log u@ stays finite.
This is implemented by getting the label at the head of the tree and discarding the rest.-}
uniform :: Prob Double
@@ -113,28 +115,48 @@
sample :: Prob a -> Meas a
sample p = Meas $ lift p
-{- | A one point measure with a given score (or weight, or mass, or likelihood), which should be a positive real number.
+{- | A one point measure with a given score (or weight, or mass, or likelihood), which should be a non-negative real number.
-A score of 0 describes impossibility. To avoid numeric issues, we encode it as @ exp(-300) @ instead.-}
+A score of 0 describes impossibility. Negative and non-finite scores raise an error.-}
score :: Double -> Meas ()
-score r = Meas $ tell $ Product $ (Exp . log) (if r==0 then exp(-300) else r)
+score r = scoreLog (Exp (log r))
+{- | 'score' with the weight given by its natural logarithm, so the weight may exceed the range of 'Double'. A logarithm of negative infinity is weight 0; a NaN or positive-infinite logarithm raises an error. -}
scoreLog :: Log Double -> Meas ()
-scoreLog r = Meas $ tell $ Product r
+scoreLog r = scoreProductLog (Product r)
+-- | 'scoreLog' with the weight wrapped in 'Product', the form in which 'Meas' accumulates weights.
scoreProductLog :: Product (Log Double) -> Meas ()
-scoreProductLog r = Meas $ tell r
+scoreProductLog (Product r) = checkedLogWeight "score" r `seq` Meas (tell (Product r))
-{- | Generate a tree with uniform random labels.
+{- | Generate a tree with labels drawn uniformly from the open interval (0,1).
This uses 'split' to split a random seed. -}
randomTree :: RandomGen g => g -> Tree
-randomTree g = let (a,g') = random g in Tree a (randomTrees g')
+randomTree g = let (a,g') = randomLabel g in Tree a (randomTrees g')
randomTrees :: RandomGen g => g -> [Tree]
randomTrees g = let (g1,g2) = split g in randomTree g1 : randomTrees g2
-{- | 'runProb' runs a probability deterministically, given a source of randomness. -}
+{- | 'runProb' runs a probability deterministically, given a source of randomness.
+
+Some results of 'runProb' hold hidden state: a counter of "LazyPPL.Distributions.Counter", a
+restaurant of 'LazyPPL.Distributions.IBP.newRestaurant', a function of
+'LazyPPL.Distributions.Memoization.generalmemoize' or 'LazyPPL.Distributions.Memoization.memrec',
+and the random functions of 'LazyPPL.Distributions.GP.wiener' and 'LazyPPL.Distributions.GP.gp'.
+Inside a model, bind such a value with @>>=@ and use it there. Outside a model, evaluate the value
+itself once, with 'Control.Exception.evaluate', before an IO action that runs more than once (the
+body of a loop or of a thread) uses it:
+
+> f <- evaluate (runProb wiener tree)
+> ys <- mapM (\x -> evaluate (f x)) xs
+
+Bound with @let@ instead, or returned inside a function from a 'Prob' computation, such a value can
+lose its state when an IO action that runs more than once uses it: GHC may copy the binding into
+the action, so that every run of the action starts from new state, and a counter then returns the
+same number to every read. Evaluating a pair or a list does not evaluate the values inside it, so
+evaluate each value with hidden state. @cabal run lazyppl:test:hidden-state-test -- known-limit@
+runs these unsupported uses. -}
runProb :: Prob a -> Tree -> a
runProb (Prob a) = a
@@ -149,29 +171,28 @@
(x, w) <- runWriterT m
rest <- helper
return $ (x, w) : rest
- newStdGen
- g <- getStdGen
+ g <- newStdGen
let rs = randomTree g
let xws = runProb helper rs
- return $ map (\(x,w) -> (x, getProduct w)) xws
+ return $ map (\(x,w) -> (x, checkedLogWeight "weightedsamples" (getProduct w))) xws
{- | Weighted importance sampling first draws n weighted samples,
and then samples a stream of results from that, regarded as an empirical distribution. Sometimes called "likelihood weighted importance sampling".
-This is a reference implementation. It will not usually be very efficient at all, but may be useful for debugging.
+This is a reference implementation. It will not usually be very efficient at all, but may be useful for debugging. Raises an error if all @n@ samples have weight 0.
-}
-wis :: Int -- ^ @n@, the number of samples to base on
+wis :: Int -- ^ @n@, the number of samples to base on, at least 1
-> Meas a -- ^ @m@, the measure to normalize
-> IO [a] -- ^ Returns a stream of samples
wis n m = do
xws <- weightedsamples m
let xws' = take n $ accumulate xws 0
let max = snd $ last xws'
- newStdGen
- g <- getStdGen
+ g <- newStdGen
let rs = (randoms g :: [Double])
- return $ map (\r -> fst $ head $ filter (\(x, w) -> w >= Exp (log r) * max) xws') rs
- where accumulate ((x, w) : xws) a = (x, w + a) : (x, w + a) : accumulate xws (w + a)
+ if max == 0 then error "wis: every sample has weight 0"
+ else return $ map (\r -> fst $ head $ filter (\(x, w) -> w > 0 && w >= Exp (log r) * max) xws') rs
+ where accumulate ((x, w) : xws) a = (x, w + a) : accumulate xws (w + a)
accumulate [] a = []
{- | Produce a stream of samples, using Metropolis Hastings simulation.
@@ -191,86 +212,98 @@
__Tip:__ if @m :: Prob a@ then use @fmap (map fst) (mh 1 $ sample m)@ to get a stream of samples from a probability distribution without conditioning.
--}
-{-- The algorithm is as follows:
- Top level: produces a stream of samples.
- * Split the random number generator in two
- * One part is used as the first seed for the simulation,
- * and one part is used for the randomness in the MH algorithm.
-
- Then, run 'step' over and over to get a stream of '(tree, result, weight)'s.
- The stream of seeds is used to produce a stream of result/weight pairs.
-
- NB There are three kinds of randomness in the step function.
- 1. The start tree 't', which is the source of randomness for simulating the
- program m to start with. This is sort-of the point in the "state space".
- 2. The randomness needed to propose a new tree ('g1')
- 3. The randomness needed to decide whether to accept or reject that ('g2')
- The tree t is an argument and result,
- but we use a state monad ('get'/'put') to deal with the other randomness '(g, g1, g2)'
-
- Steps of the 'step' function:
- 1. Randomly change some sites,
- 2. Rerun the model with the new tree, to get a new weight 'w''.
- 3. Compute the MH acceptance ratio. This is the probability of either returning the new seed or the old one.
- 4. Accept or reject the new sample based on the MH ratio.
---}
mh :: forall a.
- Double -- ^ The chance @p@ of changing any site
+ Double -- ^ The chance @p@ of changing any site, in [0,1]
-> Meas a -- ^ The unnormalized measure to sample from
-> IO [(a,Product (Log Double))] -- ^ Returns a stream of (result,weight) pairs
-mh p (Meas m) = do
- -- Top level: produce a stream of samples.
- -- Split the random number generator in two
- -- One part is used as the first seed for the simulation,
- -- and one part is used for the randomness in the MH algorithm.
- newStdGen
- g <- getStdGen
- let (g1,g2) = split g
- let t = randomTree g1
- let (x, w) = runProb (runWriterT m) t
- -- Now run step over and over to get a stream of (tree,result,weight)s.
- let (samples,_) = runState (iterateM step (t,x,w)) g2
- -- The stream of seeds is used to produce a stream of result/weight pairs.
- return $ map (\(_,x,w) -> (x,w)) samples
- {- NB There are three kinds of randomness in the step function.
- 1. The start tree 't', which is the source of randomness for simulating the
- program m to start with. This is sort-of the point in the "state space".
- 2. The randomness needed to propose a new tree ('g1')
- 3. The randomness needed to decide whether to accept or reject that ('g2')
- The tree t is an argument and result,
- but we use a state monad ('get'/'put') to deal with the other randomness '(g,g1,g2)' -}
- where step :: RandomGen g => (Tree,a,Product (Log Double)) -> State g (Tree,a,Product (Log Double))
- step (t, x, w) = do
- -- Randomly change some sites
- g <- get
- let (g1, g2) = split g
- let t' = mutateTree p g1 t
- -- Rerun the model with the new tree, to get a new
- -- weight w'.
- let (x', w') = runProb (runWriterT m) t'
- -- MH acceptance ratio. This is the probability of either
- -- returning the new seed or the old one.
- let ratio = getProduct w' / getProduct w
- let (r, g2') = random g2
- put g2'
- if r < min 1 (exp $ ln ratio) -- (trace ("-- Ratio: " ++ show ratio) ratio))
- then return (t', x', w') -- trace ("---- Weight: " ++ show w') w')
- else return (t, x, w) -- trace ("---- Weight: " ++ show w) w)
+mh p = metropolis p Nothing
+-- A label that the model never reads stays unevaluated, and its delayed
+-- mutations keep the earlier trees alive. So every 32 accepted mutateTree
+-- proposals we swap the tree for a checkpoint with the same labels, which
+-- labelAt computes from the stored seeds. The seed store in
+-- LazyPPL.Internal.History appends one 16-byte record per accepted mutateTree
+-- proposal, and its vector doubles when full. An accepted all-sites change
+-- starts a new store.
+metropolis :: Double -> Maybe Double -> Meas a -> IO [(a, Product (Log Double))]
+metropolis p refresh (Meas m)
+ -- The comparisons are false for NaN, so NaN is rejected too.
+ | not (p >= 0 && p <= 1) = error ("mh: the chance p of changing a site must be in [0,1], got " ++ show p)
+ | Just q <- refresh, not (q > 0 && q <= 1) =
+ error ("mhirreducible: the chance q of an all-sites change must be in (0,1], got " ++ show q)
+ | otherwise = do
+ g <- newStdGen
+ let (g1, g2) = split g
+ t = randomTree g1
+ (x, w) = checkedSample (runProb (runWriterT m) t)
+ history <- newHistory g1
+ let go generator tree value weight history count = unsafeInterleaveIO $ do
+ let (proposalSeed, acceptanceSeed) = split generator
+ (fresh, seed) = case refresh of
+ Nothing -> (False, proposalSeed)
+ Just q -> let (r, next) = randomLabel proposalSeed
+ in (r < q, next)
+ proposed = if fresh then randomTree seed else mutateTree p seed tree
+ (value', weight') = checkedSample (runProb (runWriterT m) proposed)
+ ratio = getProduct weight' / getProduct weight
+ (r, nextGenerator) = randomLabel acceptanceSeed
+ -- At weight 0 the ratio can be 0/0 = NaN and the test then fails,
+ -- so we accept every move until the weight is positive. r is a label, in
+ -- (0,1), as for the tree: random's 1 would reject a move of ratio 1 and
+ -- an all-sites change at q = 1.
+ if getProduct weight == 0 || r < min 1 (exp (ln ratio))
+ then do
+ (history', count') <- if fresh
+ then do h <- newHistory seed; pure (h, 0)
+ else do appendMutation history count seed; pure (history, count+1)
+ let tree' = if count' > 0 && count' `rem` 32 == 0
+ then checkpoint p history' count'
+ else proposed
+ tree' `seq` pure ()
+ rest <- go nextGenerator tree' value' weight' history' count'
+ pure ((value', weight') : rest)
+ else do
+ rest <- go nextGenerator tree value weight history count
+ pure ((value, weight) : rest)
+ go g2 t x w history 0
+ where
+ -- A product of valid weights can still overflow to a logarithm of +Infinity.
+ checkedSample (v, Product l) = (v, Product (checkedLogWeight "mh" l))
+checkpoint :: Double -> History -> Int -> Tree
+checkpoint p history count = node []
+ where
+ node path = Tree (labelAt p history count path) (children path 0)
+ children path index = node (index:path) : children path (index+1)
+
+
-- | Replace the labels of a tree randomly, with probability p
mutateTree :: forall g. RandomGen g => Double -> g -> Tree -> Tree
mutateTree p g (Tree a ts) =
- let (a',g') = (random g :: (Double,g)) in
- let (a'',g'') = random g' in
+ let (a',g') = randomLabel g in
+ let (a'',g'') = randomLabel g' in
+#ifdef LAZYPPL_BROWSER
+ -- Wasm has a 32-bit address space, so we evaluate each visited label to stop
+ -- its thunk from keeping the old trees alive.
+ let label = if a' Double -> g -> [Tree] -> [Tree]
+#ifdef LAZYPPL_BROWSER
+mutateTrees p g (t:ts) =
+ let (g1,g2) = split g
+ t' = mutateTree p g1 t
+ in t' `seq` (t' : mutateTrees p g2 ts)
+#else
mutateTrees p g (t:ts) = let (g1,g2) = split g in mutateTree p g1 t : mutateTrees p g2 ts
+#endif
-{- | Irreducible form of 'mh'. Takes @p@ like 'mh', but also @q@, which is the chance of proposing an all-sites change. Irreducibility means that, asymptotically, the sequence of samples will converge in distribution to the renormalized version of @m@.
+{- | Irreducible form of 'mh'. Takes @p@ like 'mh', but also @q@ in (0,1], which is the chance of proposing an all-sites change. Irreducibility means that, asymptotically, the sequence of samples will converge in distribution to the renormalized version of @m@, provided that the total weight of @m@ is positive and finite and that computing each weight terminates. The parts of a sample that are never demanded may be infinite and stay unevaluated. With @q = 0@ there would be no all-sites change, and so no such guarantee; @q = 0@ raises an error.
The kernel in `mh` is not formally irreducible in the usual sense, although it is an open question whether this is a problem for asymptotic convergence in any definable model. In any case, convergence is only asymptotic, and so it can be helpful to use `mhirreducible` is that in some situations.
@@ -279,49 +312,11 @@
-}
mhirreducible :: forall a.
- Double -- ^ The chance @p@ of changing any given site
- -> Double -- ^ The chance @q@ of doing an all-sites change
+ Double -- ^ The chance @p@ of changing any given site, in [0,1]
+ -> Double -- ^ The chance @q@ of doing an all-sites change, in (0,1]
-> Meas a -- ^ The unnormalized measure @m@ to sample from
-> IO [(a,Product (Log Double))] -- ^ Returns a stream of (result,weight) pairs
-mhirreducible p q (Meas m) = do
- -- Top level: produce a stream of samples.
- -- Split the random number generator in two
- -- One part is used as the first seed for the simulation,
- -- and one part is used for the randomness in the MH algorithm.
- newStdGen
- g <- getStdGen
- let (g1,g2) = split g
- let t = randomTree g1
- let (x, w) = runProb (runWriterT m) t
- -- Now run step over and over to get a stream of (tree,result,weight)s.
- let (samples,_) = runState (iterateM step (t,x,w)) g2
- -- The stream of seeds is used to produce a stream of result/weight pairs.
- return $ map (\(t,x,w) -> (x,w)) samples
- {- NB There are three kinds of randomness in the step function.
- 1. The start tree 't', which is the source of randomness for simulating the
- program m to start with. This is sort-of the point in the "state space".
- 2. The randomness needed to propose a new tree ('g1')
- 3. The randomness needed to decide whether to accept or reject that ('g2')
- The tree t is an argument and result,
- but we use a state monad ('get'/'put') to deal with the other randomness '(g,g1,g2)' -}
- where step :: RandomGen g => (Tree,a,Product (Log Double)) -> State g (Tree,a,Product (Log Double))
- step (t,x,w) = do
- -- Randomly change some sites
- g <- get
- let (g1,g2) = split g
- -- Decide whether to resample all sites (r y : every n ys
[] -> []
-
--- | Monadic iterate (inlined from monad-extras).
-iterateM :: Monad m => (a -> m a) -> a -> m [a]
-iterateM f x = do
- x' <- f x
- (x':) `liftM` iterateM f x'
diff --git a/src/LazyPPL/Distributions.hs b/src/LazyPPL/Distributions.hs
--- a/src/LazyPPL/Distributions.hs
+++ b/src/LazyPPL/Distributions.hs
@@ -1,16 +1,17 @@
-{- | Handy distributions for the `LazyPPL` library, based on the `uniform` distribution. Mostly defined using the `Statistics.Distribution` module and family.
+{- | Handy distributions for the `LazyPPL` library, based on the `uniform` distribution.
Sometimes both a distribution (type @Prob a@) and pdf (type @a -> Double@) are given. Distributions are useful for sampling, densities are used for scoring.
For more distributions, see the Statistics.Distribution in the statistics package. -}
{-# OPTIONS_GHC -Wno-unrecognised-pragmas #-}
{-# HLINT ignore "Avoid lambda using `infix`" #-}
+{-# LANGUAGE BangPatterns #-}
module LazyPPL.Distributions (
-- * Continuous distributions
normal,normalPdf,normalLogPdf,exponential,expPdf,exponentialLogPdf,
- gamma, gammaLogPdf, beta, betaLogPdf, dirichlet, dirichletLogPdf,
+ gamma, gammaLogPdf, beta, BetaLabelError(..), betaLogPdf, dirichlet, dirichletLogPdf,
uniformbounded, uniformBoundedLogPdf,
logNormalLogPdf,
-- * Discrete distributions
@@ -22,9 +23,13 @@
where
import LazyPPL (Prob,uniform)
-import Data.List (findIndex)
+import Control.Exception (Exception, throw)
+import Data.Bits (bit, shift)
+import Data.List (findIndex, zipWith5)
+import qualified Numeric.Sum as Sum
import Numeric.SpecFunctions
import Numeric.MathFunctions.Constants
+import Numeric.RootFinding (Root(..), RiddersParam(..), Tolerance(..), ridders)
{-|
[Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution)
@@ -32,11 +37,15 @@
normal :: Double -- ^ mu, mean
-> Double -- ^ sigma, standard deviation
-> Prob Double
-normal m s = do
- x <- uniform
- return $ (- invErfc (2 * x)) * (m_sqrt_2 * s) + m
+normal m s
+ | not (finite m && positiveFinite s) = error "normal: the mean must be finite and the standard deviation finite and positive"
+ | otherwise = do
+ x <- uniform
+ -- The standard normal quantile is finite on the open interval; scaling it
+ -- last avoids overflowing m_sqrt_2 * s.
+ return $ m + s * (- invErfc (2 * x) * m_sqrt_2)
normalPdf :: Double -> Double -> Double -> Double
-normalPdf m s x = exp (- ((x - m) * (x - m) / (2 * s * s)) - log (m_sqrt_2_pi * s))
+normalPdf m s x = exp (normalLogPdf m s x)
{-|
@@ -44,11 +53,13 @@
-}
exponential :: Double -- ^ lambda, rate
-> Prob Double
-exponential rate = do
- x <- uniform
- return $ - (log x / rate)
+exponential rate
+ | not (positiveFinite rate) = error "exponential: the rate must be finite and positive"
+ | otherwise = do
+ x <- uniform
+ return $ - (log x / rate)
expPdf :: Double -> Double -> Double
-expPdf rate x = exp (- (rate * x)) * rate
+expPdf rate x = exp (exponentialLogPdf rate x)
{-|
[Gamma distribution](https://en.wikipedia.org/wiki/Gamma_distribution)
@@ -56,149 +67,475 @@
gamma :: Double -- ^ k, shape
-> Double -- ^ theta, scale
-> Prob Double
-gamma a b = do
- x <- uniform
- return $ b * invIncompleteGamma a x
+gamma a b
+ | not (positiveFinite a && positiveFinite b) = error "gamma: the shape and scale must be finite and positive"
+ | otherwise = do
+ x <- uniform
+ return $ b * invIncompleteGamma a x
{-|
[Beta distribution](https://en.wikipedia.org/wiki/Beta_distribution)
+
+ Each parameter must be at least 0.001, and their sum at most 2000, unless one
+ of them is 1: Beta(a,1) and Beta(1,b) take every finite positive a and b.
+ Other parameters raise an error. For other shapes, custom tree labels below
+ @2^-53@ raise 'BetaLabelBelowGeneratorRange'; the default generator's
+ smallest label is @2^-53@. The analytic one-shape cases accept all labels in
+ @(0,1)@. On the supported rows of the reference grid, the drawn u-quantile has a distribution
+ function within @1e-9 * min u (1 - u)@ of u, or is one of the two doubles
+ next to the exact quantile.
-}
beta :: Double -- ^ alpha
-> Double -- ^ beta
-> Prob Double
-beta a b = do
- x <- uniform
- return $ invIncompleteBeta a b x
+beta a b
+ | not (positiveFinite a && positiveFinite b && (betaSupported [a, b] || a == 1 || b == 1)) =
+ error ("beta: the parameters must each be at least " ++ show betaLeast ++ " and sum to at most " ++ show betaSumMost
+ ++ ", or be 1 and a finite positive number, got " ++ show (a, b))
+ | otherwise = do
+ x <- uniform
+ return $ fst (betaQuantile a b x)
+-- | A custom tree label lies below the supported range of the general Beta
+-- sampler. The default random-tree generator never produces such a label.
+data BetaLabelError = BetaLabelBelowGeneratorRange Double deriving (Eq, Show)
+instance Exception BetaLabelError
+
{-|
[Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution)
-}
poisson :: Double -- ^ lambda, rate
-> Prob Integer
-poisson lambda = do
- x <- uniform
- let cmf = map (\x -> 1 - incompleteGamma (fromIntegral (x + 1)) lambda) [0,1..]
- let (Just n) = findIndex (> x) cmf
- return $ fromIntegral n
+poisson lambda
+ | not (lambda >= 0 && finite lambda) = error "poisson: the rate must be finite and non-negative"
+ | otherwise = do
+ x <- uniform
+ -- Compare the smaller tail directly, so subtracting it from 1 cannot erase it.
+ let tails = map (\n -> incompleteGamma (fromIntegral (n + 1)) lambda) [0,1..]
+ cumulative = map Sum.kbn $ drop 1 $ scanl Sum.add Sum.zero (map (exp . poissonLogPmf lambda) [0..])
+ (Just n) = if x >= 0.5 then findIndex (< 1 - x) tails else findIndex (> x) cumulative
+ return $ fromIntegral n
poissonPdf :: Double -> Integer -> Double
-poissonPdf rate n = let result = exp(-rate) * rate ^^ fromIntegral n / factorial (fromIntegral n) in
- if isInfinite result || isNaN result then exp (-rate + fromIntegral n * log rate - logGamma (fromIntegral (n+1))) else result
+poissonPdf rate n = exp (poissonLogPmf rate n)
{-|
[Dirichlet distribution](https://en.wikipedia.org/wiki/Dirichlet_distribution)
+
+ Drawn by stick breaking: the first coordinate is Beta(a1, a2 + ... + ak), each
+ later coordinate takes a Beta(ai, a(i+1) + ... + ak) share of the stick that
+ remains, and the last coordinate is the rest. This is the Dirichlet law, and no
+ coordinate is NaN even when the concentrations are so small that Gamma draws
+ underflow to 0.
+
+ Each concentration must be at least 0.001, and their sum at most 2000; other
+ concentrations raise an error. Each share is a quantile of its Beta law, with
+ the accuracy of 'beta'.
-}
dirichlet :: [Double] -- ^ vector of alphas; length is dimension
-> Prob[Double]
-dirichlet as = do
- xs <- mapM (\a -> gamma a 1) as
- let s = Prelude.sum xs
- let ys = map (/ s) xs
- return ys
+dirichlet as
+ | not (betaSupported as) = error ("dirichlet: the concentrations must each be at least " ++ show betaLeast
+ ++ " and sum to at most " ++ show betaSumMost ++ ", got " ++ show as)
+ | otherwise = breakStick 1 as (drop 1 (scanr (+) 0 as))
+ where
+ breakStick rest [_] _ = return [rest]
+ breakStick rest (a : rest') (later : laters) = do
+ u <- uniform
+ let (v, w) = betaQuantile a later u
+ xs <- breakStick (rest * w) rest' laters
+ return (rest * v : xs)
+ breakStick _ _ _ = return []
+-- | The supported parameters of dirichlet, and of beta apart from Beta(a,1) and
+-- Beta(1,b) (see betaQuantile): each at least betaLeast, with a sum of at most
+-- betaSumMost. Every stick of dirichlet as is then a Beta law whose two
+-- parameters lie in the domain, as their sum is a suffix sum of as.
+betaSupported :: [Double] -> Bool
+betaSupported as = all (>= betaLeast) as && sum as <= betaSumMost
+
+-- The bounds are measured. test/reference/reference.py writes, with mpmath, the
+-- rows of test/reference/beta-quantiles.tsv: Beta(a,b) at a label u, with the
+-- interval of doubles that a supported draw may take (see betaQuantile), or a
+-- rejection marker when both samplers reject the input. It covers 23
+-- parameters from 1e-10 to 1e10 at 13 labels from 1e-300 to 1 - 2^-53, and for
+-- the corners of the bounds, the doubles just outside them and random pairs
+-- inside. On 1 Nov 2026 (math-functions 0.3.4.4, GHC 9.14.1), before the label
+-- restriction and with the former first-order intervals, the bounds removed,
+-- and each row run in a process killed after 2 s, beta a b and dirichlet [a,b]
+-- landed in the intervals at all 2057 rows inside the shape bounds,
+-- the slowest in 0.35 s (Beta(1999.999,0.001) at 1e-300). Outside, rows landed
+-- outside the intervals where both parameters exceed 3000, where math-functions
+-- computes I by quadrature (Beta(1e4,1e4) at 1e-300), and ran past 2 s with a
+-- parameter of 1e-5 or below (Beta(1000,1e-5) at 2^-53). Draws with a parameter
+-- near 0.001 slow down as the sum grows: Beta(1e4,0.001) took 1.6 s at 1e-20.
+betaLeast, betaSumMost :: Double
+betaLeast = 1e-3
+betaSumMost = 2000
+
+-- | The u-quantile x of Beta(a,b), with 1 - x, for parameters in the supported
+-- domain. The quantile is a root of the regularized incomplete Beta function I
+-- of math-functions: its inverse, invIncompleteBeta (0.3.4.4), starts from
+-- guesses that fail outside a few regimes (its issues #29, #44 and #47), and
+-- returns 0 for Beta(1, 0.001) at 1/4, whose quantile is 1 - 0.75^1000. Of x
+-- and 1 - x, the one at most 1/2 is solved for, through
+-- I_x(a,b) = 1 - I_(1-x)(b,a), so that each keeps its relative precision even
+-- when the other rounds to 1. Both probabilities u and 1 - u are passed on:
+-- 1 - u is exact when u >= 1/2, and u when it is the smaller.
+betaQuantile :: Double -> Double -> Double -> (Double, Double)
+betaQuantile a b u
+ -- Beta(a,1) has the distribution function x^a and Beta(1,b) has 1 - (1-x)^b,
+ -- so their u-quantiles are u^(1/a) and 1 - (1-u)^(1/b). With t = log u / a or
+ -- t = log1p (-u) / b, exp t and -expm1 t retain the smaller share without
+ -- subtracting it from 1. If division overflows to negative infinity, that
+ -- smaller share rounds to 0.
+ | b == 1 = let t = log u / a in (exp t, - expm1 t)
+ | a == 1 = let t = log1p (- u) / b in (- expm1 t, exp t)
+ | u < encodeFloat 1 (-53) = throw (BetaLabelBelowGeneratorRange u)
+ | u <= incompleteBeta a b 0.5 = let x = smallQuantile a b u (1 - u) in (x, 1 - x)
+ | otherwise = let y = smallQuantile b a (1 - u) u in (1 - y, y)
+
+-- | The v-quantile z <= 1/2 of Beta(p,q), given I_(1/2)(p,q) >= v and w = 1 - v:
+-- the root of a residual in t = log z, between the least positive double and
+-- 1/2, by math-functions' Ridders method, which keeps the bracket. Solving for
+-- log z keeps full relative precision for quantiles far below 1, such as
+-- 0.75^1000. The residual compares the logarithms of the smaller of the two
+-- probabilities: for w at least 2^-19, log I_z(p,q) with log v; below, the
+-- upper tail 1 - I_z(p,q) = I_(1-z)(q,p) with log w. Near 1, I_z(p,q) is a
+-- double within 2^-53 of 1 - w, so comparing it with v would leave the root
+-- anywhere on a plateau of relative width 2^-53 / w in w, at most 2^-34 for w
+-- at least 2^-19. Where a probability underflows, its logarithm is held at that
+-- of the least positive double, so the residual stays finite and keeps its
+-- sign; a probability that is not a number in [0,1] raises an error. A quantile
+-- below the least positive double is 0, the nearer double or the one below it.
+-- The search stops within one ulp of t (AbsTol 0) and may take 200 steps,
+-- twice math-functions' default, because the bracket spans the whole double
+-- range on the log scale; a search that does not converge raises an error
+-- rather than returning a guess.
+smallQuantile :: Double -> Double -> Double -> Double -> Double
+smallQuantile p q v w
+ | f hi <= 0 = 0.5
+ | f lo > 0 = 0
+ | otherwise = case ridders RiddersParam { riddersMaxIter = 200, riddersTol = AbsTol 0 } (lo, hi) f of
+ Root t -> exp t
+ _ -> error ("beta: no quantile found for parameters " ++ show (p, q) ++ " and probability " ++ show v)
+ where
+ f t | w < encodeFloat 1 (-19) = logProbability w - logProbability (incompleteBeta q p (1 - exp t))
+ | otherwise = logProbability (incompleteBeta p q (exp t)) - logProbability v
+ logProbability c
+ | c >= 0 && c <= 1 = log (max leastPositive c)
+ | otherwise = error ("beta: the incomplete Beta function gave " ++ show c ++ " for parameters " ++ show (p, q))
+ leastPositive = encodeFloat 1 (-1074)
+ lo = log leastPositive
+ hi = log 0.5
+
-- | [Continuous uniform distribution on a bounded interval](https://en.wikipedia.org/wiki/Continuous_uniform_distribution)
uniformbounded :: Double -- ^ lower
-> Double -- ^ upper
-> Prob Double
-uniformbounded lower upper = do
- x <- uniform
- return $ (upper - lower) * x + lower
+uniformbounded lower upper
+ | not (finite lower && finite upper && lower < upper) = error "uniformbounded: the bounds must be finite, with lower < upper"
+ | otherwise = do
+ x <- uniform
+ -- The width overflows for finite bounds of opposite signs near the largest
+ -- double; the weighted sum of the bounds does not (random 1.3.1's
+ -- scaleFloating handles an infinite width the same way).
+ return $ if isInfinite (upper - lower) then x * upper + (1 - x) * lower else (upper - lower) * x + lower
-- | [Bernoulli distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution)
bernoulli :: Double -- ^ bias
-> Prob Bool
-bernoulli r = do
- x <- uniform
- return $ x < r
+bernoulli r
+ | not (probability r) = error "bernoulli: the probability must be in [0,1]"
+ | otherwise = do
+ x <- uniform
+ return $ x < r
{-|
[Discrete uniform distribution](https://en.wikipedia.org/wiki/Discrete_uniform_distribution) on [0, ..., n-1]
-}
uniformdiscrete :: Int -- ^ n
-> Prob Int
-uniformdiscrete n =
- do
- let upper = fromIntegral n
- r <- uniformbounded 0 upper
- return $ floor r
+uniformdiscrete n
+ | n < 1 = error "uniformdiscrete: n must be at least 1"
+ | otherwise = do
+ let upper = fromIntegral n
+ r <- uniformbounded 0 upper
+ return $ floor r
{-| [Categorical distribution](https://www.google.com/search?client=safari&rls=en&q=categorical+distribution&ie=UTF-8&oe=UTF-8): Takes a list of k numbers that sum to 1,
- and returns a random number between 0 and (k-1), weighted accordingly -}
+ and returns a random number between 0 and (k-1), weighted accordingly.
+ Only the probabilities up to the chosen index are read, so the list may be infinite, and each one read must be in [0,1]. -}
categorical :: [Double] -> Prob Int
categorical xs = do
r <- uniform
- case findIndex (>r) $ tail $ scanl (+) 0 xs of
- Just i -> return i
- Nothing -> error "categorical: probabilities do not sum to 1"
+ return $! pickCategorical r xs
+pickCategorical :: Double -> [Double] -> Int
+pickCategorical r = pick 0 Sum.zero 0 Nothing
+ where
+ -- The upper bounds below are added as exact integers, in units of
+ -- 2^-1075 (half the smallest positive Double). For example, 0.5 is
+ -- 2^1074 units and the smallest positive Double is 2 units. Every
+ -- bound from upperWeight is a whole number of these units, so their
+ -- sum never rounds. Probability 1 is 2^1075 units. The probabilities
+ -- themselves are added as Doubles (a compensated sum), which can
+ -- round, as in the example further down.
+ limit = bit 1075 :: Integer
+ -- The largest exact probability that rounds to the Double q:
+ -- q plus half the gap to the next Double, in the same units.
+ upperWeight q
+ | q == 0 = 1
+ | otherwise = let (mantissa, exponent) = decodeFloat q
+ in shift mantissa (exponent + 1075) + bit (max 0 (exponent + 1074))
+ pick !i !total !upper !previous (q:qs)
+ | not (probability q) = error "categorical: probabilities must be in [0,1]"
+ | q > 0 && Sum.kbn next > r = i
+ | otherwise = pick (i + 1) next (min limit (upper + upperWeight q))
+ -- Capped at 1, so an infinite list cannot grow it.
+ (if q > 0 then Just i else previous) qs
+ where next = Sum.add total q
+ -- The list ended before the running sum passed r. Valid input can do
+ -- this. For example, with 49 weights of 1/49:
+ -- stored Double: 0.02040816326530612 (just below 1/49)
+ -- sum of all 49: 0.9999999999999999 (not 1)
+ -- draw r: 0.9999999999999999 (the largest draw below 1)
+ -- No running sum passes r. Each true weight is at most upperWeight of
+ -- its Double, so if the true weights sum to 1, the upper bounds sum to
+ -- at least 1. If they do, the gap between the sum and 1 (1.1e-16 in
+ -- the example) comes from rounding: return the last index with
+ -- positive weight (index 48 in the example). If not, the weights sum
+ -- to less than 1 by more than rounding can explain.
+ pick _ _ !upper (Just i) []
+ | upper >= limit = i
+ | otherwise = error "categorical: probabilities do not sum to 1"
+ pick _ _ _ Nothing [] = error "categorical: probabilities do not sum to 1"
+
{-| Returns an infinite stream of samples from the given distribution. --}
iid :: Prob a -> Prob [a]
iid p = do r <- p; rs <- iid p; return $ r : rs
normalLogPdf :: Double -> Double -> Double -> Double
-normalLogPdf mu sigma x =
- let d = x - mu
- in -0.5 * log (2 * pi) - log sigma - (d * d) / (2 * sigma * sigma)
+normalLogPdf mu sigma x
+ | not (finite mu && positiveFinite sigma) = error "normalLogPdf: the mean must be finite and the standard deviation finite and positive"
+ | otherwise = -0.5 * log (2 * pi) - log sigma - (0.5 * z) * z
+ where
+ d = x - mu
+ -- Standardise before squaring, so (x - mu)^2 and sigma^2 cannot overflow or
+ -- underflow on their own. x - mu overflows only for huge x and mu of opposite
+ -- signs, and then x/sigma - mu/sigma loses nothing.
+ z | isInfinite d && not (isInfinite x) = x / sigma - mu / sigma
+ | otherwise = d / sigma
exponentialLogPdf :: Double -> Double -> Double
exponentialLogPdf rate x
- | x < 0 = -1e300
+ | rate <= 0 || isNaN rate || isInfinite rate = error "exponentialLogPdf: rate must be finite and positive"
+ | x < 0 = m_neg_inf
| otherwise = log rate - rate * x
gammaLogPdf :: Double -> Double -> Double -> Double
gammaLogPdf k theta x
- | x <= 0 = -1e300
- | otherwise = (k - 1) * log x - x / theta - k * log theta - logGamma k
+ | not (positiveFinite k && positiveFinite theta) = error "gammaLogPdf: the shape and scale must be finite and positive"
+ | x <= 0 || isInfinite x = m_neg_inf
+ | isInfinite lambda && isInfinite relativeRate = m_neg_inf
+ | isInfinite lambda = k * log1pmx relativeRate - stirlingError k
+ - m_ln_sqrt_2_pi - 0.5 * log k + log k - log x
+ -- The density is the Poisson mass of lambda = x/theta at k - 1, divided by
+ -- theta (at k when k < 1, times k/x), so the Poisson kernel applies.
+ | lambda == 0 = (k - 1) * log x - k * log theta - logGamma k
+ | k == 1 = - lambda - log theta
+ | k > 1 = poissonKernel (k - 1) lambda (nearResidual k (lambda - (k - 1)) (rate - (toRational k - 1))) - log theta
+ | otherwise = poissonKernel k lambda (nearResidual k (lambda - k) (rate - toRational k)) + log k - log x
+ where
+ lambda = x / theta
+ rate = toRational x / toRational theta
+ relativeRate = fromRational ((rate - toRational k) / toRational k)
+-- | Log density of the Beta distribution, or negative infinity outside (0,1).
+-- Parameters that are not positive, or whose sum is not finite, raise an error.
betaLogPdf :: Double -> Double -> Double -> Double
betaLogPdf a b x
- | x <= 0 || x >= 1 = -1e300
- | otherwise = (a - 1) * log x + (b - 1) * log (1 - x) - logBeta a b
+ | not (positiveFinite a && positiveFinite b && positiveFinite (a + b)) = error "betaLogPdf: both parameters must be positive, with a finite sum"
+ | x <= 0 || x >= 1 = m_neg_inf
+ | otherwise = betaKernel a b x (1 - x) (log x) (log1p (-x)) (toRational x)
bernoulliLogPmf :: Double -> Bool -> Double
-bernoulliLogPmf p True = log (max 1e-300 p)
-bernoulliLogPmf p False = log (max 1e-300 (1 - p))
+bernoulliLogPmf p outcome
+ | isNaN p || p < 0 || p > 1 = error "bernoulliLogPmf: probability must be in [0,1]"
+ | outcome = log p
+ | otherwise = log1p (-p)
+-- | Log-probability of the k-th category. Only the probabilities at indices
+-- 0..k are inspected, so an infinite probability stream, as the Dirichlet
+-- process supplies, is answered after reading its needed prefix.
categoricalLogPmf :: [Double] -> Int -> Double
categoricalLogPmf probs k
- | k < 0 || k >= length probs = -1e300
- | otherwise = log (max 1e-300 (probs !! k))
+ | k < 0 = m_neg_inf
+ | otherwise = case drop k probs of
+ [] -> m_neg_inf
+ (p:_)
+ | any (\q -> isNaN q || q < 0 || q > 1) (take (k + 1) probs) ->
+ error "categoricalLogPmf: probabilities must be in [0,1]"
+ | otherwise -> log p
uniformBoundedLogPdf :: Double -> Double -> Double -> Double
uniformBoundedLogPdf lower upper x
- | x < lower || x > upper = -1e300
+ | not (finite lower && finite upper && lower < upper) = error "uniformBoundedLogPdf: the bounds must be finite, with lower < upper"
+ | x < lower || x > upper = m_neg_inf
+ | isInfinite (upper - lower) = -(log (upper / 2 - lower / 2) + log 2)
| otherwise = -log (upper - lower)
+-- | Log density of the Dirichlet distribution, or negative infinity outside
+-- the open simplex. The coordinates must sum to 1 within
+-- gamma_n = n*u/(1-n*u), with n the dimension and u = 2^-53, which bounds the
+-- rounding error of the coordinates and their sum. Concentrations that are not
+-- positive, or whose sum is not finite, raise an error.
dirichletLogPdf :: [Double] -> [Double] -> Double
dirichletLogPdf alphas xs
- | length alphas /= length xs = -1e300
- | any (<= 0) xs = -1e300
- | otherwise =
- let lnB = sum (map logGamma alphas) - logGamma (sum alphas)
- in sum (zipWith (\a x -> (a - 1) * log x) alphas xs) - lnB
+ | null xs || length alphas /= length xs = m_neg_inf
+ | not (all positiveFinite alphas && positiveFinite (sum alphas)) = error "dirichletLogPdf: concentrations must be positive, with a finite sum"
+ | any (\x -> x <= 0 || isNaN x || isInfinite x) xs = m_neg_inf
+ | abs (sum xs - 1) > roundingBound = m_neg_inf
+ | otherwise = sum (zipWith5 factor alphas laters xs rests (drop 1 rests))
+ where
+ nu = fromIntegral (length xs) * encodeFloat 1 (negate (floatDigits (0 :: Double)))
+ roundingBound = nu / (1 - nu)
+ -- With r_i = x_i + ... + x_k, the share x_i/r_i of the remaining stick is
+ -- Beta(a_i, a_(i+1) + ... + a_k), independently (the aggregation property),
+ -- so the density is the product of those Beta densities over the Jacobian
+ -- r_i of x_i = r_i * (x_i/r_i). Suffix sums form 1 - x_i/r_i without cancelling.
+ rests = scanr1 (+) xs
+ laters = drop 1 (scanr (+) 0 alphas)
+ factor a later x r next =
+ betaKernel a later (x / r) (next / r) (log x - log r) (log next - log r) (toRational x / toRational r) - log r
logNormalLogPdf :: Double -> Double -> Double -> Double
logNormalLogPdf mu sigma x
- | x <= 0 = -1e300
- | otherwise =
- let lx = log x
- d = lx - mu
- in -0.5 * log (2 * pi) - log sigma - lx - (d * d) / (2 * sigma * sigma)
+ | not (finite mu && positiveFinite sigma) = error "logNormalLogPdf: the mean must be finite and the standard deviation finite and positive"
+ | x <= 0 = m_neg_inf
+ | otherwise = normalLogPdf mu sigma (log x) - log x
binomialLogPmf :: Int -> Double -> Int -> Double
binomialLogPmf n p k
- | k < 0 || k > n = -1e300
- | p <= 0 && k > 0 = -1e300
- | p >= 1 && k < n = -1e300
- | otherwise =
- logGamma (fromIntegral n + 1)
- - logGamma (fromIntegral k + 1)
- - logGamma (fromIntegral (n - k) + 1)
- + fromIntegral k * log (max 1e-300 p)
- + fromIntegral (n - k) * log (max 1e-300 (1 - p))
+ | n < 0 || isNaN p || p < 0 || p > 1 = error "binomialLogPmf: n must be non-negative and probability in [0,1]"
+ | k < 0 || k > n = m_neg_inf
+ | p == 0 = if k == 0 then 0 else m_neg_inf
+ | p == 1 = if k == n then 0 else m_neg_inf
+ | k == 0 = fromIntegral n * log1p (-p)
+ | k == n = fromIntegral n * log p
+ | otherwise = binomialKernel (fromIntegral k) (fromIntegral (n - k)) p (1 - p)
+ (nearResidual (fromIntegral n) (fromIntegral n * p - fromIntegral k) (fromIntegral n * toRational p - fromIntegral k))
poissonLogPmf :: Double -> Integer -> Double
poissonLogPmf lambda k
- | k < 0 = -1e300
- | otherwise = fromIntegral k * log lambda - lambda - logGamma (fromIntegral k + 1)
+ | lambda < 0 || isNaN lambda || isInfinite lambda = error "poissonLogPmf: rate must be finite and non-negative"
+ | k < 0 = m_neg_inf
+ -- At rate 0 the formula below gives NaN for k = 0 (0 * log 0), as does
+ -- statistics-0.16.5.0.
+ | lambda == 0 = if k == 0 then 0 else m_neg_inf
+ | k == 0 = - lambda
+ | isInfinite count = largeCountPoisson lambda k
+ | otherwise = poissonKernel (fromIntegral k) lambda (countResidual lambda k)
+ where count = fromInteger k :: Double
+
+-- For a count above Double's finite range, keep one quarter of it in range.
+-- The exact Integer residual retains central counts that round to the rate.
+largeCountPoisson :: Double -> Integer -> Double
+largeCountPoisson lambda k
+ | isInfinite quarter = m_neg_inf
+ | relative > -0.5 = quarter * (4 * log1pmx relative) - correction
+ | otherwise = - quarter * (4 * (logRatio - 1) + lambda / quarter) - correction
+ where
+ quarter = fromInteger (k `div` 4)
+ relative = fromRational ((toRational lambda - fromInteger k) / fromInteger k)
+ logCount = log quarter + log 4
+ ratio = (lambda / quarter) / 4
+ logRatio | ratio >= m_tiny = - log ratio
+ | otherwise = logCount - log lambda
+ correction = stirlingError quarter / 4 + m_ln_sqrt_2_pi + 0.5 * logCount
+
+-- | lambda - k, rounded once. A count above 2^53 can differ from the nearest
+-- double by more than the spread of the distribution (10^32 and the double 1e32
+-- differ by 5.4e15, half a standard deviation), so its difference from lambda is
+-- formed exactly.
+countResidual :: Double -> Integer -> Double
+countResidual lambda k
+ | k < 2 ^ (53 :: Int) = lambda - fromInteger k
+ | otherwise = fromRational (toRational lambda - fromInteger k)
+
+-- Loader's saddle-point forms (C. Loader, "Fast and accurate computation of
+-- binomial probabilities", 2000). log(n!) is the Stirling formula plus
+-- stirlingError n (from math-functions), and each power is a deviance bd0, so
+-- no two large terms cancel, as they did in sums of logGamma values. Logarithms
+-- of products are taken as sums of logarithms, so no product leaves the range.
+
+-- | The logarithm of lambda^m e^(-lambda) / Gamma(m+1), for real m > 0 and
+-- lambda > 0, given lambda - m as d.
+poissonKernel :: Double -> Double -> Double -> Double
+poissonKernel m lambda d = - stirlingError m - bd0 m lambda d - m_ln_sqrt_2_pi - 0.5 * log m
+
+-- | The logarithm of Gamma(m+l+1) / (Gamma(m+1) Gamma(l+1)) p^m q^l, for real
+-- m, l > 0 and q = 1 - p given separately, given np - m as d; nq - l is -d.
+binomialKernel :: Double -> Double -> Double -> Double -> Double -> Double
+binomialKernel m l p q d =
+ stirlingError n - stirlingError m - stirlingError l
+ - bd0 m (n * p) d - bd0 l (n * q) (- d)
+ + 0.5 * (log n - log m - log l) - m_ln_sqrt_2_pi
+ where n = m + l
+
+-- | The residual np - x of bd0 near the centre, rounded once, for a count or
+-- shape `size`. Rounding np, a product or quotient of the inputs, by a
+-- relative 2^-53 moves the deviance by at most |np - x| 2^-53, which stays
+-- below 1e-12 of the log density while size is below 2^25: so below, the
+-- residual comes from Double arithmetic (`fast`), and above, from the exact
+-- rational value of the same expression. At gammaLogPdf 1e300 1e-300 1, the
+-- rounded x / theta is one ulp, 1.5e284, from 1e300, and the deviance 1.1e268
+-- that it gives is 3.7 times the true 3.0e267.
+nearResidual :: Double -> Double -> Rational -> Double
+nearResidual size fast exact
+ | size < 33554432 = fast
+ | otherwise = fromRational exact
+
+-- | The deviance x log(x/np) + np - x, for positive finite x and np, given
+-- d = np - x. With r = np/x it is -x (log r - (r - 1)) = -x log1pmx (r - 1).
+-- For r in [1/2, 2], r - 1 = d/x, and math-functions' log1pmx is accurate near
+-- 0, where the deviance is small; between doubles, np - x is exact there. Elsewhere
+-- |log(x/np)| >= log 2 and the deviance is at least 0.15 max(x, np): it is
+-- x (log(x/np) - 1) + np, whose two terms sum in magnitude to at most 13.4
+-- times its value, and neither exceeds it in magnitude by more than np, so
+-- neither overflows where the deviance does not (x log(x/np) itself is 2.1e308
+-- at x = 1e308, np = 1.25e307, where the deviance is 1.2e308). It takes
+-- log x - log np when x/np is not a normal double. math-functions' own bd0
+-- (0.3.4.4) forms x/np and x + np: it gives Infinity at x = 1, np = 1e-320, and
+-- at x = np = 1e308 its series starts from Infinity * 0 and never ends.
+bd0 :: Double -> Double -> Double -> Double
+bd0 x np d
+ | x / 2 <= np && np / 2 <= x = - x * log1pmx (d / x)
+ | otherwise = x * (logRatio - 1) + np
+ where
+ ratio = x / np
+ logRatio | ratio >= m_tiny && ratio <= m_huge = log ratio
+ | otherwise = log x - log np
+
+-- | The log density of Beta(a,b) at p, with q = 1 - p and both logarithms given,
+-- so that none of the four is formed by cancellation, and p also exactly.
+betaKernel :: Double -> Double -> Double -> Double -> Double -> Double -> Rational -> Double
+betaKernel a b p q logP logQ exactP
+ -- (a + b - 1) times the binomial mass of a - 1 successes in a + b - 2 trials.
+ -- With both counts at least 1 the kernel's own Stirling terms do not cancel.
+ | a >= 2 && b >= 2 = log (a + b - 1) + binomialKernel (a - 1) (b - 1) p q
+ (nearResidual (a + b) ((a + b - 2) * p - (a - 1)) ((toRational a + toRational b - 2) * exactP - (toRational a - 1)))
+ -- Below 2 the direct form has no large terms to cancel; logBeta corrects a
+ -- large argument with Stirling terms.
+ | otherwise = (a - 1) * logP + (b - 1) * logQ - logBeta a b
+
+-- Parameter domains. A comparison with NaN is false, so each test rejects NaN.
+positiveFinite :: Double -> Bool
+positiveFinite x = x > 0 && x < m_pos_inf
+
+finite :: Double -> Bool
+finite x = x > m_neg_inf && x < m_pos_inf
+
+probability :: Double -> Bool
+probability p = p >= 0 && p <= 1
diff --git a/src/LazyPPL/Distributions/Counter.hs b/src/LazyPPL/Distributions/Counter.hs
--- a/src/LazyPPL/Distributions/Counter.hs
+++ b/src/LazyPPL/Distributions/Counter.hs
@@ -1,4 +1,3 @@
-{-# LANGUAGE BangPatterns #-}
module LazyPPL.Distributions.Counter (Counter,newCounter,readAndIncrement) where
import LazyPPL
@@ -18,16 +17,21 @@
data Counter = C (IORef Int)
+-- | A counter that starts at 0. A counter holds hidden state: see 'LazyPPL.runProb' for its use outside a model.
newCounter :: Prob Counter
+-- Both operations stay out of the caller's code. Inlined there, full laziness
+-- (on at GHC's default -O) gave a counter created and read inside one Prob
+-- computation the number 0 at every read (test/HiddenState.hs, -O1 and -O2).
+{-# NOINLINE newCounter #-}
newCounter = do r <- uniform
return $ C $ unsafePerformIO $ newIORef (round (r-r))
+-- | The counter's next number: 0, then 1, and so on.
readAndIncrement :: Counter -> Prob Int
+{-# NOINLINE readAndIncrement #-}
readAndIncrement (C ref) = do
r <- uniform
- return $ unsafePerformIO $ do
- !i <- readIORef ref
- () <- writeIORef ref (i + 1 + round (r - r))
- return i
+ -- One atomic step, so two threads never receive the same number.
+ return $ unsafePerformIO $ atomicModifyIORef' ref (\i -> (i + 1 + round (r - r), i))
diff --git a/src/LazyPPL/Distributions/DirichletP.hs b/src/LazyPPL/Distributions/DirichletP.hs
--- a/src/LazyPPL/Distributions/DirichletP.hs
+++ b/src/LazyPPL/Distributions/DirichletP.hs
@@ -18,8 +18,8 @@
-- * Random distribution interface
dp) where
-import Data.List
-import Data.Maybe
+import Data.Coerce (coerce)
+import qualified Numeric.Sum as Sum
import LazyPPL
import LazyPPL.Distributions
import LazyPPL.Distributions.Memoization (MonadMemo)
@@ -33,35 +33,41 @@
{-| A customer enters the restaurant and is assigned a table. -}
newCustomer :: Restaurant -> Prob Table
-newCustomer (R restaurant) =
- do
- r <- uniform
- return $ T $ fromJust $ findIndex (> r) (scanl1 (+) restaurant)
+-- Coercing the newtype preserves categorical's random-tree allocation.
+newCustomer (R restaurant) = coerce (categorical restaurant)
-{-| Create a new restaurant with concentration parameter alpha. -}
+{-| Create a new restaurant with concentration parameter alpha, which may be any finite positive number; other values raise an error. -}
newRestaurant :: Double -- ^ Concentration parameter, alpha
-> Prob Restaurant
-newRestaurant alpha = do
- sticks <- stickBreaking alpha 0
- return $ R sticks
+newRestaurant alpha
+ | not (concentration alpha) = error ("newRestaurant: the concentration alpha must be finite and positive, got " ++ show alpha)
+ | otherwise = do
+ sticks <- stickBreaking alpha Sum.zero
+ return $ R sticks
+-- | Whether alpha is a concentration that beta 1 alpha takes: finite and positive.
+concentration :: Double -> Bool
+concentration alpha = alpha > 0 && not (isInfinite alpha)
+
{- | Stick breaking breaks the unit interval into an
infinite number of parts (lazily) --}
-stickBreaking :: Double -> Double -> Prob [Double]
+stickBreaking :: Double -> Sum.KBNSum -> Prob [Double]
stickBreaking alpha lower =
do
r <- beta 1 alpha
- let v = r * (1 - lower)
- vs <- stickBreaking alpha (lower + v)
+ let v = r * (1 - Sum.kbn lower)
+ vs <- stickBreaking alpha (Sum.add lower v)
return (v : vs)
-{-| [Dirichlet Process](https://en.wikipedia.org/wiki/Dirichlet_process) as a random distribution. -}
+{-| [Dirichlet Process](https://en.wikipedia.org/wiki/Dirichlet_process) as a random distribution. The concentration alpha may be any finite positive number; other values raise an error. -}
dp :: Double -- ^ Concentration parameter, alpha
-> Prob a -- ^ Base distribution
-> Prob (Prob a)
-dp alpha p = do
- xs <- iid p
- vs <- stickBreaking alpha 0
- return $ do
- r <- uniform
- return $ xs !! fromJust (findIndex (> r) (scanl1 (+) vs))
+dp alpha p
+ | not (concentration alpha) = error ("dp: the concentration alpha must be finite and positive, got " ++ show alpha)
+ | otherwise = do
+ xs <- iid p
+ vs <- stickBreaking alpha Sum.zero
+ -- Prob's fmap adds a tree split. Map this result on the same tree so the
+ -- shared selector also preserves the existing fixed-seed distribution.
+ return $ Prob $ \tree -> xs !! runProb (categorical vs) tree
diff --git a/src/LazyPPL/Distributions/GP.hs b/src/LazyPPL/Distributions/GP.hs
--- a/src/LazyPPL/Distributions/GP.hs
+++ b/src/LazyPPL/Distributions/GP.hs
@@ -9,15 +9,13 @@
-}
-module LazyPPL.Distributions.GP (wiener, gp) where
+module LazyPPL.Distributions.GP (wiener, gp, GPError(..)) where
import LazyPPL (Prob,Tree(Tree),Prob(Prob),runProb)
import LazyPPL.Distributions (normal,iid)
-import Numeric.LinearAlgebra hiding (step,size,find)
-import Numeric.LinearAlgebra.Data hiding (step,size,find)
-import Data.List (find)
-import Data.Map (empty,lookup,insert,size,keys,elems,Map)
-import Data.IORef
+import LazyPPL.Distributions.GP.Pure (GPError(..), emptyGPState, sampleGP)
+import qualified Data.Map
+import Control.Concurrent.MVar (modifyMVar, newMVar)
import System.IO.Unsafe
import Prelude hiding ((<>))
@@ -25,71 +23,78 @@
This is a random function.
-The Wiener process is implemented by using a Brownian bridge and a hidden memo table. -}
+The Wiener process is implemented by using a Brownian bridge and a hidden memo table.
+A query reads the table, draws and stores under one lock, so concurrent queries
+condition on each other; if a query raises an error, the table is unchanged.
+The returned function holds hidden state: see 'LazyPPL.runProb' for its use outside a model. -}
wiener :: Prob (Double -> Double)
-wiener = Prob $ \(Tree r gs) ->
- unsafePerformIO $ do
- ref <- newIORef Data.Map.empty
- modifyIORef' ref (Data.Map.insert 0 0)
- return $ \x -> unsafePerformIO $ do
- table <- readIORef ref
- case Data.Map.lookup x table of
- Just y -> do {return y}
- Nothing -> do let lower = do {l <- findMaxLower x (keys table) ;
- v <- Data.Map.lookup l table ; return (l,v) }
- let upper = do {u <- find (> x) (keys table) ;
- v <- Data.Map.lookup u table ; return (u,v) }
- let m = bridge lower x upper
- let y = runProb m (gs !! (1 + size table))
- modifyIORef' ref (Data.Map.insert x y)
- return y
+wiener = Prob $ \(Tree _ gs) ->
+ unsafePerformIO $ do
+ ref <- newMVar (Data.Map.singleton 0 0)
+ return $ \x -> unsafePerformIO $ modifyMVar ref $ \table ->
+ if isNaN x || isInfinite x then error ("wiener: the time must be finite, got " ++ show x) else
+ case Data.Map.lookup x table of
+ Just y -> return (table, y)
+ Nothing -> do
+ let m = bridge (Data.Map.lookupLT x table) x (Data.Map.lookupGT x table)
+ y = runProb m (gs !! (1 + Data.Map.size table))
+ y `seq` return (Data.Map.insert x y table, y)
+-- | The value at time y given its nearest neighbours in the table: the
+-- conditional mean plus the conditional standard deviation times a standard
+-- normal draw. It reads the same label of the tree as normal mean sd would, so
+-- each point keeps its subtree.
bridge :: Maybe (Double,Double) -> Double -> Maybe (Double,Double) -> Prob Double
--- not needed since the table is always initialized with (0, 0)
--- bridge Nothing y Nothing = if y==0 then return 0 else normal 0 (sqrt y)
-bridge (Just (x,x')) y Nothing = normal x' (sqrt (y-x))
-bridge Nothing y (Just (z,z')) = normal z' (sqrt (z-y))
-bridge (Just (x,x')) y (Just (z,z')) = normal (x' + ((y-x)*(z'-x')/(z-x))) (sqrt ((z-y)*(y-x)/(z-x)))
-
-findMaxLower :: Double -> [Double] -> Maybe Double
-findMaxLower d [] = Nothing
-findMaxLower d (x:xs) = let y = findMaxLower d xs in
- case y of
- Nothing -> if x < d then Just x else Nothing
- Just m -> do
- if x > m && x < d then Just x else Just m
-
-
-
+bridge lower y upper = Prob $ \t -> mean + sd * runProb (normal 0 1) t
+ where
+ (mean, sd) = case (lower, upper) of
+ (Just (x,x'), Nothing) -> (x', sqrt (y - x))
+ (Nothing, Just (z,z')) -> (z', sqrt (z - y))
+ -- Between x < y < z the variance is left * right / width, with left = y - x,
+ -- right = z - y and width = z - x. As sqrt (min left right) times
+ -- sqrt (max left right / width), whose second factor lies between sqrt (1/2)
+ -- and 1, the standard deviation is representable whenever its true value is, even
+ -- where the variance or left / width is not. The mean moves from the nearer
+ -- neighbour by that neighbour's share of the width, at most 1/2.
+ (Just (x,x'), Just (z,z')) ->
+ let left = y - x
+ right = z - y
+ width = z - x
+ in ( if left <= right then x' + (left / width) * (z' - x') else z' - (right / width) * (z' - x')
+ , sqrt (min left right) * sqrt (max left right / width) )
+ -- The table always holds the point 0.
+ (Nothing, Nothing) -> error "wiener: empty table"
{-| [Gaussian process](https://en.wikipedia.org/wiki/Gaussian_process) with given covariance function.
This is a random function.
The function is defined by using linear algebra and a hidden memo table.
-Although it uses hidden state, it is still safe, i.e. statistically commutative and discardable. --}
-gp :: (Double -> Double -> Double) -- ^ Covariance function
- -> Prob (Double -> Double) -- ^ Returns a random function
-gp cov = do ns <- iid $ normal 0 1
- return $ unsafePerformIO $ do
- ref <- newIORef Data.Map.empty
- modifyIORef' ref (Data.Map.insert 0 0)
- return $ \x -> unsafePerformIO $ do
- table <- readIORef ref
- case Data.Map.lookup x table of
- Just y -> do {return y}
- Nothing -> do let y = step cov table x $ ns !! (1 + size table)
- modifyIORef' ref (Data.Map.insert x y)
- return y
-step :: (Double -> Double -> Double) -> Data.Map.Map Double Double -> Double -> Double -> Double
-step cov table x seed =
- let sig11 = cov x x
- sig12 = matrix (size table) [cov x b | b <-keys table]
- sig21 = matrix 1 [cov a x | a <-keys table]
- sig22 = matrix (size table) [cov a b | a <- keys table , b <-keys table]
- regCoeff = sig12 <> (pinvTol 1E8 sig22)
- mu = (regCoeff <> (matrix 1 $ elems table)) `atIndex` (0,0)
- var = sig11 - ((regCoeff <> sig21) `atIndex` (0,0)) in
- mu + seed * (sqrt $ if var > -0.01 then (abs var) else error (show var))
+At distinct points the values have covariance
+@C(x,y) = cov(x,y) + 2^-33 * cov(x,x) * [x == y]@, and a repeated query returns
+the same value. The covariance function must be symmetric, which is not
+checked. A point with zero variance returns 0, and its covariance with every
+other queried point must be 0.
+A covariance value that is NaN or infinite raises 'GPCovarianceNotFinite'. Any
+other failure, including an overflow or a NaN in the
+[Cholesky factorization](https://en.wikipedia.org/wiki/Cholesky_decomposition)
+of @C@, raises 'GPCovarianceNotPositiveDefinite'. A query reads the table,
+draws and stores under one lock, so concurrent queries condition on each other;
+a failed query leaves the table unchanged. The returned function holds hidden
+state: see 'LazyPPL.runProb' for its use outside a model. -}
+gp :: (Double -> Double -> Double) -- ^ Covariance function
+ -> Prob (Double -> Double) -- ^ Returns a random function
+gp cov = do
+ ns <- iid (normal 0 1)
+ return $ unsafePerformIO $ do
+ ref <- newMVar (Data.Map.empty, emptyGPState)
+ return $ \x -> unsafePerformIO $ modifyMVar ref $ \(table, state) ->
+ case Data.Map.lookup x table of
+ Just value -> return ((table, state), value)
+ Nothing -> do
+ let seed = ns !! (1 + Data.Map.size table)
+ (value, next) = sampleGP cov state x seed
+ -- Force sampleGP's checks inside modifyMVar, so a failed query changes nothing.
+ value `seq` return ((Data.Map.insert x value table, next), value)
diff --git a/src/LazyPPL/Distributions/GP/Pure.hs b/src/LazyPPL/Distributions/GP/Pure.hs
new file mode 100644
--- /dev/null
+++ b/src/LazyPPL/Distributions/GP/Pure.hs
@@ -0,0 +1,69 @@
+{-# LANGUAGE BangPatterns #-}
+{-# OPTIONS_GHC -O2 #-}
+module LazyPPL.Distributions.GP.Pure (GPError(..), GPState, emptyGPState, sampleGP) where
+
+import Control.Exception (Exception, throw)
+import Data.List (foldl')
+
+-- | A 'GPError' holds the point that could not be sampled, and its constructor says why.
+data GPError
+ = GPCovarianceNotFinite Double
+ | GPCovarianceNotPositiveDefinite Double
+ deriving (Eq, Show)
+
+instance Exception GPError
+
+-- Rows, and each row's entries for earlier points, are newest first.
+data Row = Row !Double !Double !Double [Double]
+
+-- | The checked rows of the [Cholesky factor](https://en.wikipedia.org/wiki/Cholesky_decomposition),
+-- and the zero-variance points, which have no row.
+data GPState = GPState [Row] [Double]
+
+-- | Nothing sampled yet.
+emptyGPState :: GPState
+emptyGPState = GPState [] []
+
+-- | @sampleGP cov state x seed@ samples at a new point @x@ from an independent
+-- standard normal draw @seed@. The values have covariance
+-- @C(x,y) = cov(x,y) + 2^-33 * cov(x,x) * [x == y]@, and zero-variance points
+-- get 0.
+sampleGP :: (Double -> Double -> Double) -> GPState -> Double -> Double -> (Double, GPState)
+sampleGP cov (GPState rows zeros) x seed
+ | not (finite prior && finite diagonal) || any nonfiniteCross cross =
+ throw (GPCovarianceNotFinite x)
+ | prior < 0 || any invalidZero cross =
+ throw (GPCovarianceNotPositiveDefinite x)
+ | prior == 0 = (0, GPState rows (x:zeros))
+ | pivot <= 0 || isNaN pivot =
+ throw (GPCovarianceNotPositiveDefinite x)
+ | otherwise = (sum (zipWith (*) lower seeds) + pivot*seed,
+ GPState (Row x seed pivot lower:rows) zeros)
+ where
+ prior = cov x x
+ diagonal = prior + nugget * prior
+ cross = [(False,cov x a,cov a x) | Row a _ _ _ <- rows] ++
+ [(True,cov x a,cov a x) | a <- zeros]
+ nonfiniteCross (_,a,b) = not (finite a && finite b)
+ invalidZero (zero,a,b) = (prior == 0 || zero) && (a /= 0 || b /= 0)
+ seeds = [z | Row _ z _ _ <- rows]
+ lower = foldl' append [] (reverse rows)
+ append previous (Row a _ d coefficients) =
+ let !value = (cov x a - sum (zipWith (*) coefficients previous)) / d
+ in value:previous
+ pivot = sqrt (diagonal - sum (map (\a -> a*a) lower))
+
+-- The nugget is a fixed fraction of each point's variance, so C does not
+-- depend on the query order. It stops the factorization failing on nearly
+-- dependent rows, as for an RBF kernel on a fine grid.
+nugget :: Double
+nugget = 2 ** (-33)
+
+-- `lower` and `pivot` in sampleGP are one Cholesky-Banachiewicz step, forward
+-- substitution against the earlier rows followed by a positive pivot. `matrix`
+-- 0.3.6.4 has no incremental Cholesky API, so its `cholDecomp` would factor the
+-- whole matrix again at each new point, O(m^3) work per query against O(m^2)
+-- here, with m points already queried.
+
+finite :: Double -> Bool
+finite value = not (isNaN value || isInfinite value)
diff --git a/src/LazyPPL/Distributions/IBP.hs b/src/LazyPPL/Distributions/IBP.hs
--- a/src/LazyPPL/Distributions/IBP.hs
+++ b/src/LazyPPL/Distributions/IBP.hs
@@ -5,12 +5,15 @@
{-|
An implementation of the Indian buffet process by [Griffiths and Ghahramani](https://papers.nips.cc/paper_files/paper/2005/file/2ef35a8b78b572a47f56846acbeef5d3-Paper.pdf).
-We are using abstract types to hide the implementation details, inspired by [Exchangeable Random Processes and Data Abstraction](https://www.cs.ox.ac.uk/people/hongseok.yang/paper/pps17a.pdf).
+The restaurant types hide their implementation state, inspired by [Exchangeable Random Processes and Data Abstraction](https://www.cs.ox.ac.uk/people/hongseok.yang/paper/pps17a.pdf). The `Dish` constructor exposes the column index used by feature-matrix examples.
Illustration: [Feature extraction example](https://lazyppl-team.github.io/AdditiveClusteringDemo.html).
-}
-module LazyPPL.Distributions.IBP where
+module LazyPPL.Distributions.IBP
+ ( Restaurant, Dish(..), newCustomer, newRestaurant, ibp
+ , RestaurantS, DishS, newCustomerS, newRestaurantS
+ ) where
import LazyPPL
import LazyPPL.Distributions
@@ -31,6 +34,8 @@
return [ D k | k <- [0..(length (matrix!!i) - 1)], matrix!!i!!k ]
+-- | A restaurant of the Indian buffet process with concentration alpha. Each 'newCustomer' takes the next row
+-- of its matrix, so a restaurant holds hidden state: see 'LazyPPL.runProb' for its use outside a model.
newRestaurant :: Double -> Prob Restaurant
newRestaurant alpha = do
r <- uniform
@@ -46,9 +51,8 @@
existingDishes <- mapM (\m -> bernoulli (fromIntegral m / i)) features
let newFeatures = zipWith (\ a b -> if b then a + 1 else a) features existingDishes
nNewDishes <- fromIntegral <$> poisson (alpha / i)
- let fixZero = if features == [] && nNewDishes == 0 then 1 else nNewDishes
- let newRow = existingDishes ++ take fixZero (repeat True)
- rest <- matrix alpha (index + 1) (newFeatures ++ take fixZero (repeat 1))
+ let newRow = existingDishes ++ take nNewDishes (repeat True)
+ rest <- matrix alpha (index + 1) (newFeatures ++ take nNewDishes (repeat 1))
return $ newRow : rest
-- the distribution on matrices
@@ -57,35 +61,53 @@
-{--
-Another possible implementation of the indian buffet process
-which uses a truncated stickbreaking construction.
-It is only an approximation to the true IBP, but doesn't need IO.
-
-See also
-Stick-breaking Construction for the Indian Buffet Process
-Teh, Gorur, Ghahramani. AISTATS 2007.
-
-A stochastic programming perspective on nonparametric Bayes
-Daniel M. Roy, Vikash Mansinghka, Noah Goodman, and Joshua Tenenbaum
-ICML Workshop on Nonparametric Bayesian, 2008.
---}
+-- | A restaurant for the approximation 'newRestaurantS' to the Indian buffet process. It holds the
+-- finite list of sticks, the probabilities with which a customer takes its dishes.
newtype RestaurantS = RS [Double]
+-- | A dish of a 'RestaurantS' restaurant: one dish for each stick, in the order of the sticks.
newtype DishS = DS Int deriving (Eq,Ord,Show)
+-- | A new customer of a 'RestaurantS' restaurant. It takes each dish independently, with the
+-- probability given by the dish's stick.
newCustomerS :: RestaurantS -> Prob [DishS]
newCustomerS (RS rs) = do
fs <- mapM bernoulli rs
return $ map DS $ findIndices id fs
+{-| An /approximation/ to 'newRestaurant': the Indian buffet process built by stick breaking and cut
+short. The concentration alpha may be any finite positive number; other values raise an error. Its
+law is not the law of 'newRestaurant'; the difference is bounded as follows.
+
+The full construction draws @nu_1, nu_2, ...@ independently from @Beta(alpha, 1)@ and forms the sticks
+@r_1 = nu_1@, @r_2 = nu_1 * nu_2@, and so on. A customer takes the @k@-th dish with probability @r_k@,
+independently for each @k@.
+
+This function generates the sticks in order and stops at the first one below 0.01. That stick is kept
+as the last one: every stick before it is at least 0.01, and it is below 0.01. The later sticks are
+not generated.
+
+The later sticks are the error. Given the last kept stick @r@, they are @r * nu'_1@,
+@r * nu'_1 * nu'_2@, and so on, where the @nu'_j@ are further independent draws from @Beta(alpha, 1)@,
+whose mean is @alpha / (alpha + 1)@. Each later stick is the probability of one dish, so the expected
+number of dishes that this function leaves out of one customer's row, compared with the exact process,
+is the expected sum of the later sticks:
+
+> r * (alpha / (alpha + 1)) / (1 - alpha / (alpha + 1)) = alpha * r
+
+As @r < 0.01@, that is fewer than @0.01 * alpha@ dishes. The bound grows with @alpha@: it is below 0.1
+when @alpha = 10@ and below 10 when @alpha = 1000@. Use 'newRestaurant' when that error matters.
+
+See also /Stick-breaking Construction for the Indian Buffet Process/, Teh, Gorur and Ghahramani,
+AISTATS 2007, and /A stochastic programming perspective on nonparametric Bayes/, Roy, Mansinghka,
+Goodman and Tenenbaum, ICML Workshop on Nonparametric Bayesian, 2008.
+-}
newRestaurantS :: Double -> Prob RestaurantS
-newRestaurantS a = RS <$> stickScale 1
+newRestaurantS a
+ | not (a > 0 && not (isInfinite a)) = error ("newRestaurantS: the concentration alpha must be finite and positive, got " ++ show a)
+ | otherwise = RS <$> stickScale 1
where stickScale p = do r' <- beta a 1
let r = p * r'
-- Truncate when the probabilities are getting small
rs <- if r < 0.01 then return [] else stickScale r
return $ r : rs
-
-
-
diff --git a/src/LazyPPL/Distributions/Memoization.hs b/src/LazyPPL/Distributions/Memoization.hs
--- a/src/LazyPPL/Distributions/Memoization.hs
+++ b/src/LazyPPL/Distributions/Memoization.hs
@@ -24,13 +24,15 @@
module LazyPPL.Distributions.Memoization (MonadMemo, memoize, generalmemoize, memrec) where
-import Control.Monad
-import Data.IORef
-import Data.Map (empty, insert, keys, lookup, size)
+import Data.IORef (atomicModifyIORef', newIORef)
+import Data.Map (empty, insert, lookup, size)
+import Data.Tuple (Solo(MkSolo))
import LazyPPL
import System.IO.Unsafe
-{-| Type class for memoizable argument types @a@ under a monad @m@ -}
+{-| Type class for memoizable argument types @a@ under a monad @m@.
+
+The default 'memoize', for an enumerable @a@, samples an infinite trie of results inside @m@, so it returns only in a lazy monad such as 'Prob'. A strict monad such as 'Maybe' or 'IO' would never return the memo function, so give no instance for one. -}
class (Monad m) => MonadMemo m a where
memoize :: (a -> m b) -> m (a -> b)
default memoize :: (Enum a) => (a -> m b) -> m (a -> b)
@@ -40,18 +42,25 @@
return $ \x -> look t (fromEnum x)
{-- Basic trie based integer-indexed memo table.
- NB Currently ignores negative integers --}
+ Negative indices are rejected. --}
data BinTree a = Branch a (BinTree a) (BinTree a)
ini :: (Monad m) => Int -> (Int -> m a) -> m (BinTree a)
ini n f = do x <- f n; l <- ini (2 * n + 1) f; r <- ini (2 * n + 2) f; return $ Branch x l r
look :: BinTree a -> Int -> a
-look (Branch x l r) 0 = x
-look (Branch _ l r) n = if even n then look r (n `div` 2) else look l (n `div` 2)
+look tree n
+ | n < 0 = error "memoize: negative indices are not supported"
+ | otherwise = follow tree (reverse (path (toInteger n + 1)))
+ where
+ path :: Integer -> [Bool]
+ path 1 = []
+ path k = odd k : path (k `div` 2)
+ follow (Branch x _ _) [] = x
+ follow (Branch _ l r) (goRight : rest) = follow (if goRight then r else l) rest
-{-| Implementation for enumerable types using tries -}
-instance (Monad m) => MonadMemo m Int
+{-| Implementation for enumerable types using tries, in the lazy 'Prob' monad. Indices must be non-negative. -}
+instance MonadMemo Prob Int
{-| Implementation for pair types using currying -}
instance (Monad m, MonadMemo m a, MonadMemo m b) => MonadMemo m (a, b) where
@@ -63,40 +72,35 @@
a table of calls that have already been made.
If @a@ is finite, we could just sample all values of @a@ in advance
and avoid unsafePerformIO. If @a@ is enumerable, we can use the trie method.
+
+ The returned function holds hidden state: see 'LazyPPL.runProb' for its use outside a model.
-}
generalmemoize :: Ord a => (a -> Prob b) -> Prob (a -> b)
-generalmemoize f = Prob $ \(Tree _ gs) ->
- unsafePerformIO $ do
- ref <- newIORef Data.Map.empty
- return $ \x -> unsafePerformIO $ do
- m <- liftM (Data.Map.lookup x) (readIORef ref)
- case m of
- Just y -> return y
- Nothing -> do
- n <- readIORef ref
- let y = runProb (f x) (gs !! (1 + size n))
- modifyIORef' ref (Data.Map.insert x y)
- return y
+generalmemoize f = memrec (const f)
-{-- Stochastic memoization for recursive functions.
+{-| Stochastic memoization for recursive functions.
Applying 'memoize' to a recursively defined function only memoizes at the
top-level: recursive calls are calls to the non-memoized function.
'memrec' is an alternative implementation which resolves recursion and
memoization at the same time, so that recursive calls are also memoized.
---}
-memrec :: Ord a => Show a => ((a -> b) -> (a -> Prob b)) -> Prob (a -> b)
+
+ The returned function holds hidden state: see 'LazyPPL.runProb' for its use outside a model.
+-}
+memrec :: Ord a => ((a -> b) -> (a -> Prob b)) -> Prob (a -> b)
memrec f =
Prob $ \(Tree _ gs) ->
unsafePerformIO $ do
ref <- newIORef Data.Map.empty
+ -- One atomic step looks up the key and, for a new key, stores its result
+ -- with the next unused subtree, so two threads never share a subtree or
+ -- lose an entry. The result stays lazy (MkSolo keeps atomicModifyIORef'
+ -- from forcing it), so a recursive call evaluates after the step.
let memoized_fixpoint = \x -> unsafePerformIO $ do
- m <- liftM (Data.Map.lookup x) (readIORef ref)
- case m of
- Just y -> return y
- Nothing -> do
- n <- readIORef ref
- let fix = f memoized_fixpoint
- let y = runProb (fix x) (gs !! (1 + size n))
- modifyIORef' ref (Data.Map.insert x y)
- return y
+ MkSolo y <- atomicModifyIORef' ref $ \table ->
+ case Data.Map.lookup x table of
+ Just y -> (table, MkSolo y)
+ Nothing ->
+ let y = runProb (f memoized_fixpoint x) (gs !! (1 + size table))
+ in (Data.Map.insert x y table, MkSolo y)
+ return y
return memoized_fixpoint
diff --git a/src/LazyPPL/Distributions/Mondrian.hs b/src/LazyPPL/Distributions/Mondrian.hs
new file mode 100644
--- /dev/null
+++ b/src/LazyPPL/Distributions/Mondrian.hs
@@ -0,0 +1,149 @@
+{-# LANGUAGE ExtendedDefaultRules, ScopedTypeVariables, BangPatterns #-}
+{-# OPTIONS_GHC -Wno-unrecognised-pragmas #-}
+{-# HLINT ignore "Use newtype instead of data" #-}
+
+module LazyPPL.Distributions.Mondrian where
+
+import LazyPPL
+import Data.IORef
+-- import Debug.Trace
+import LazyPPL.Distributions
+import LazyPPL.Distributions.Counter
+import System.IO.Unsafe
+import Data.Map (Map,empty,lookup,insert,size,keys,findWithDefault,fromList,(!))
+-- import Text.RawString.QQ
+-- import qualified Data.List
+-- import Control.DeepSeq
+
+
+-- | From Roy-Teh "The Mondrian Process", NIPS 2009
+
+-- | 'oneDimMondrian' corresponds to a Poisson point process on [low, high] with uniform intensity
+oneDimMondrian :: Double -> (Double, Double) -> Prob [Double]
+oneDimMondrian budget (low, high) = do
+ cutCost <- exponential (high - low)
+ if budget < cutCost
+ then return []
+ else do
+ let remaining = budget - cutCost
+ cut <- uniformbounded low high
+ leftcuts <- oneDimMondrian remaining (low, cut)
+ rightcuts <- oneDimMondrian remaining (cut, high)
+ return $ leftcuts ++ [cut] ++ rightcuts
+
+-- | Some data types. Intuitively, Matrix is a type of 2D exchangeable arrays, and 'Mondrian Double' would be the type of block-structured graphons.
+
+data Mondrian a
+ = Block a [(Double, Double)] -- ^ args: and intervals making up the block
+ | Partition Int Double [(Double, Double)] (Mondrian a) (Mondrian a) -- ^ args: dimension cutPosition intervals (hypercube over which the Mondrian is defined) subtree1
+ deriving (Eq, Show)
+
+newtype Row = Row Int -- ^ in [0, 1]
+
+newtype Col = Col Int
+
+data MatrixIO = MatrixIO Counter Counter [[Bool]]
+
+lookup :: MatrixIO -> Row -> Col -> Bool
+lookup (MatrixIO _ _ matrix) (Row r) (Col c) = matrix !! r !! c
+
+newRow :: MatrixIO -> Prob Row
+newRow (MatrixIO c _ _) = readAndIncrement c >>= (return . Row)
+
+newCol :: MatrixIO -> Prob Col
+newCol (MatrixIO _ c _) = readAndIncrement c >>= (return . Col)
+
+data Matrix = Matrix [[Bool]]
+
+{- | Generates a random Mondrian tree: a random block partition of the rectangle [a1, b1] × ⋯ × [an, bn]
+where each block has an associated random draw from the base distribution.
+-}
+randomMondrian :: Prob a -> Double -> [(Double, Double)] -> Prob (Mondrian a)
+randomMondrian base budget abs = do
+ let lengths = map (\(a, b) -> b - a) abs
+ let sumLengths = sum lengths
+ cutCost <- exponential sumLengths
+ if budget < cutCost
+ then do p <- base; return $ Block p abs
+ else do
+ let remaining = budget - cutCost
+ dim <- categorical $ map (/sumLengths) lengths -- if dim is true then cut is perpendicular to (a_d, b_d)
+ let (a_d, b_d) = abs !! dim
+ cut <- uniformbounded a_d b_d
+ leftMondrian <- randomMondrian base remaining
+ $ zipWith (\ab i -> if i == dim then (a_d, cut) else ab) abs [0 ..]
+ rightMondrian <- randomMondrian base remaining
+ $ zipWith (\ab i -> if i == dim then (cut, b_d) else ab) abs [0 ..]
+ return $ Partition dim cut abs leftMondrian rightMondrian
+
+-- | Given a Mondrian tree and two data points, apply a function on the biais of the corresponding block.
+applyOnBiaisFromMondrian2D :: Mondrian Double -> Double -> Double -> (Double -> a) -> a
+applyOnBiaisFromMondrian2D mondrian x y f = do
+ case mondrian of
+ Block p _ -> f p
+ Partition !dim !cut _ left right ->
+ let !xy = case dim of {0 -> x; _ -> y}
+ !cond = xy < cut
+ !lr = if cond then left else right in
+ applyOnBiaisFromMondrian2D lr x y f
+
+-- | Given a Mondrian tree and two data points, sample an 'edge'.
+sampleFromMondrian2D :: Mondrian Double -> Double -> Double -> Prob Bool
+sampleFromMondrian2D mondrian x y = applyOnBiaisFromMondrian2D mondrian x y bernoulli
+
+-- | Probability of getting a given truth value 'val' at a (r, c) pair from a Mondrian.
+likelihoodFromMondrian2D :: Mondrian Double -> Double -> Double -> Bool -> Double
+likelihoodFromMondrian2D mondrian x y val =
+ let !f = if val then id else (1-) in
+ applyOnBiaisFromMondrian2D mondrian x y f
+
+{- | Sample a block structure over [0, 1]^2 from the Mondrian process,
+and use it to generate an infinite exchangeable array.
+-}
+sampleRelationFromRandomMondrian2D :: Prob Double -> Double -> Prob Matrix
+sampleRelationFromRandomMondrian2D base budget = do
+ mondrian <- randomMondrian base budget [(0, 1), (0, 1)]
+ sampleRelationFromMondrian2D mondrian
+
+sampleRelationFromMondrian2D :: Mondrian Double -> Prob Matrix
+sampleRelationFromMondrian2D mondrian = do
+ rs <- iid uniform
+ cs <- iid uniform
+ matrix <- mapM (\r -> mapM (sampleFromMondrian2D mondrian r) cs) rs
+ return $ Matrix matrix
+
+sampleMapRelationFromMondrian2D :: Mondrian Double -> Int -> Prob (Map (Double, Double) Bool)
+sampleMapRelationFromMondrian2D mondrian size = do
+ xs <- iid uniform
+ ys <- iid uniform
+ matrix <- mapM (\x ->
+ mapM (\y -> (do b <- sampleFromMondrian2D mondrian x y; return ((x, y), b))) (take size xs))
+ (take size ys)
+ return $ fromList $ concat matrix
+
+sampleMatrixRelationFromMondrian2D :: Mondrian Double -> Prob Matrix
+sampleMatrixRelationFromMondrian2D mondrian = do
+ rs <- iid uniform
+ cs <- iid uniform
+ matrix <- mapM (\r ->
+ mapM (sampleFromMondrian2D mondrian r) cs) rs
+ return $ Matrix matrix
+
+
+data BinTree a = Leaf | Node a (BinTree a) (BinTree a)
+ deriving (Eq, Show)
+
+{- | Create a Mondrian tree over a hypercube `abs` from a tree of (dimension, cut) given in the chronological order, and a map associating to each block its bernoulli biais.
+-}
+mondrianFromCuts :: [(Double, Double)] -> BinTree (Int, Double) -> Map [(Double, Double)] Double -> Double -> Mondrian Double
+mondrianFromCuts abs dcs biais defaultBiais =
+ case dcs of
+ Leaf -> Block (findWithDefault defaultBiais abs biais) abs
+ Node (dim, cut) left right ->
+ Partition dim cut abs
+ (mondrianFromCuts leftAbs left biais defaultBiais)
+ (mondrianFromCuts rightAbs right biais defaultBiais)
+ where
+ leftAbs = zipWith (\ab i -> if i == dim then (a0, cut) else ab) abs [0 ..]
+ rightAbs = zipWith (\ab i -> if i == dim then (cut, b0) else ab) abs [0 ..]
+ (a0, b0) = abs !! dim
diff --git a/src/LazyPPL/Internal/History.hs b/src/LazyPPL/Internal/History.hs
new file mode 100644
--- /dev/null
+++ b/src/LazyPPL/Internal/History.hs
@@ -0,0 +1,89 @@
+{-# LANGUAGE BangPatterns #-}
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+
+module LazyPPL.Internal.History
+ (History, newHistory, appendMutation, labelAt) where
+
+import Control.Concurrent.MVar
+import Control.Monad (unless)
+import Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.Vector.Unboxed.Mutable as Vector
+import Data.Word (Word64)
+import System.IO.Unsafe (unsafePerformIO)
+import LazyPPL.Internal.Sampler (randomLabel)
+import System.Random (SeedGen(fromSeed64, toSeed64), StdGen, split)
+
+-- A record holds a StdGen as the two Word64 values (16 bytes) of its public
+-- SeedGen instance in `random` 1.3. Stored unboxed, the records contain no
+-- pointers, so the garbage collector skips them, and the `vector` package does
+-- the allocation and copying.
+data Cursor = Cursor !(Vector.IOVector (Word64,Word64)) !Int
+data History = History !StdGen !(MVar (Maybe Cursor))
+
+newHistory :: StdGen -> IO History
+newHistory seed = History seed <$> newMVar Nothing
+
+appendMutation :: History -> Int -> StdGen -> IO ()
+appendMutation (History _ lock) index generator =
+ modifyMVar_ lock $ \old -> do
+ Cursor records count <- case old of
+ Nothing -> do
+ unless (index == 0) (ioError (userError "missing MH history"))
+ records <- Vector.new 1
+ pure (Cursor records 0)
+ Just cursor -> pure cursor
+ unless (count == index) (ioError (userError "out-of-order MH history"))
+ unless (index < (maxBound :: Int))
+ (ioError (userError "MH history exceeds addressable record count"))
+ let capacity = Vector.length records
+ grown <- if index < capacity
+ then pure records
+ -- Grow by doubling, without overflowing Int.
+ else Vector.grow records (min capacity (maxBound-capacity))
+ case toSeed64 generator of
+ seed :| [gamma] -> Vector.write grown index (seed,gamma)
+ _ -> ioError (userError "StdGen seed is not 128 bits")
+ pure (Just (Cursor grown (index+1)))
+
+-- `labelAt p history count path` is the label at `path` (deepest index first)
+-- after the first `count` mutations, where a mutation replaces each label
+-- independently with probability `p`.
+--
+-- readMVar waits for an append in progress, so those `count` records are
+-- initialized. Later appends write only after them, and growth copies them to
+-- a new vector. A reader's vector and count therefore stay valid.
+{-# NOINLINE labelAt #-}
+labelAt :: Double -> History -> Int -> [Int] -> Double
+labelAt p (History initial lock) count reversedPath = unsafePerformIO $
+ readMVar lock >>= \cursor -> do
+ let path = reverse reversedPath
+ original = fst (randomLabel (atPath 1 path initial))
+ if count == 0 || not (p > 0)
+ then pure original
+ else case cursor of
+ Nothing -> ioError (userError "missing MH checkpoint records")
+ Just (Cursor records written) -> do
+ unless (count <= written) (ioError (userError "incomplete MH checkpoint"))
+ let search :: Int -> IO Double
+ search !index
+ | index < 0 = pure original
+ | otherwise = do
+ (seed,gamma) <- Vector.read records index
+ let generator = atPath 2 path (fromSeed64 (seed :| [gamma]))
+ (decision, next) = randomLabel generator
+ if decision < p
+ then pure (fst (randomLabel next))
+ else search (index-1)
+ value <- search (count-1)
+ value `seq` pure value
+
+-- At each node, randomTree draws once before splitting its seed and mutateTree
+-- draws twice.
+atPath :: Int -> [Int] -> StdGen -> StdGen
+atPath _ [] generator = generator
+atPath draws (index:path) generator = atPath draws path (child index (advance draws generator))
+ where
+ advance 0 g = g
+ advance n g = advance (n-1) (snd (randomLabel g))
+ child 0 g = fst (split g)
+ child n g = child (n-1) (snd (split g))
diff --git a/src/LazyPPL/Internal/Sampler.hs b/src/LazyPPL/Internal/Sampler.hs
new file mode 100644
--- /dev/null
+++ b/src/LazyPPL/Internal/Sampler.hs
@@ -0,0 +1,32 @@
+{- | Rules that every sampler of the random tree shares: how a label is drawn,
+and which weights inference accepts. This module is internal. -}
+module LazyPPL.Internal.Sampler (randomLabel, checkedLogWeight) where
+
+import Data.Bits (complement, shiftR)
+import Numeric.Log (Log, ln)
+import System.Random (RandomGen, genWord64)
+
+{- | A label of the random tree, uniform on the open interval (0,1).
+
+@random@ from random 1.3.1 gives a 'Double' in the closed interval [0,1], and
+an inverse transform fails at an end point: @-log 0@ is infinite, and a
+Bernoulli(1) coin compares a label of 1 with 1. With k the 52 high bits of
+the complement of one 'Data.Word.Word64', this gives @(k + 1/2) / 2^52@, in
+@[2^-53, 1 - 2^-53]@. It reads one word, as @random@ does, so every node of a
+tree gets the same generator as before; both labels decrease as the word
+grows, and they differ by at most @2^-52@. -}
+randomLabel :: RandomGen g => g -> (Double, g)
+randomLabel g =
+ let (w, g') = genWord64 g
+ in (encodeFloat (2 * toInteger (complement w `shiftR` 12) + 1) (-53), g')
+
+{- | A weight is a non-negative real number, kept as its logarithm so that it
+may exceed the range of 'Double': the logarithm is finite, or negative
+infinity for weight 0. A NaN or positive-infinite logarithm raises an error
+that names the caller. -}
+checkedLogWeight :: String -> Log Double -> Log Double
+checkedLogWeight caller w
+ | isNaN l || (l > 0 && isInfinite l) =
+ error (caller ++ ": the weight must be non-negative and finite, but its logarithm is " ++ show l)
+ | otherwise = w
+ where l = ln w
diff --git a/test/Api.hs b/test/Api.hs
new file mode 100644
--- /dev/null
+++ b/test/Api.hs
@@ -0,0 +1,91 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Main where
+
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import StrictMemo (strictMemoChecks)
+import LazyPPL.Distributions
+import LazyPPL.Distributions.GP (gp, wiener)
+import LazyPPL.Distributions.IBP (ibp)
+import LazyPPL.Distributions.Memoization (memoize, memrec)
+import System.Environment (getArgs, getExecutablePath)
+import System.Exit (exitFailure, exitSuccess)
+
+import Api.Support
+import Api.Domains
+import Api.Distributions
+import Api.Processes
+import Api.Inference
+
+-- Accuracy of samplers, densities and processes at extreme parameters.
+accuracyChecks :: IO [Bool]
+accuracyChecks = concat <$> sequence [dirichletSamplingChecks, lawChecks, gridLawChecks, betaQuantileChecks False, dirichletStickChecks, densityChecks, wienerBridgeChecks, numericChecks, processChecks, strictMemoChecks]
+
+-- Sampler and density domains: tree labels, log weights, chances, distribution
+-- parameters and the concentrations of the nonparametric priors.
+domainChecks :: IO [Bool]
+domainChecks = concat <$> sequence [endpointChecks, logWeightChecks, chanceChecks, parameterChecks, betaLabelChecks, concentrationChecks]
+
+main :: IO ()
+main = do
+ args <- getArgs
+ case args of
+ ["categorical-boundary"] -> categoricalBoundaryChecks >>= finish
+ ["categorical-properties"] -> categoricalPropertyChecks >>= finish
+ ["dirichlet-stick-rounding"] -> dirichletStickRoundingChecks >>= finish
+ ["poisson-boundary"] -> poissonBoundaryChecks >>= finish
+ ["memo"] -> memoStochasticChecks >>= finish
+ ["mondrian"] -> mondrianChecks >>= finish
+ ["zero-weight"] -> zeroAcceptanceChecks >>= finish
+ ["domains"] -> domainChecks >>= finish
+ ["accuracy"] -> accuracyChecks >>= finish
+ ["laws"] -> concat <$> sequence [dirichletSamplingChecks, lawChecks, gridLawChecks, betaQuantileChecks False, dirichletStickChecks] >>= finish
+ ["quantiles"] -> betaQuantileChecks False >>= finish
+ ["quantiles-inside"] -> betaQuantileChecks True >>= finish
+ ["sticks"] -> dirichletStickChecks >>= finish
+ ["grid-laws"] -> gridLawChecks >>= finish
+ ["densities"] -> densityChecks >>= finish
+ ["density-rows"] -> densityRowsChild
+ ["wiener"] -> concat <$> sequence [wienerBridgeChecks, processChecks] >>= finish
+ ["posterior"] -> posteriorChecks >>= finish
+ _ -> do
+ additional <- concat <$> sequence [categoricalBoundaryChecks, categoricalPropertyChecks, dirichletStickRoundingChecks, poissonBoundaryChecks, densityBoundaryChecks, lazyCategoricalChecks, memoStochasticChecks, mondrianChecks, recursiveMemoCheck, dirichletProcessChecks, zeroAcceptanceChecks, domainChecks, accuracyChecks, posteriorChecks]
+ if args == ["additional"] then finish additional else existingChecks additional
+
+finish :: [Bool] -> IO ()
+finish results = do
+ let passed = length (filter id results)
+ total = length results
+ putStrLn $ show passed ++ "/" ++ show total ++ " checks passed."
+ if passed == total then exitSuccess else exitFailure
+
+existingChecks :: [Bool] -> IO ()
+existingChecks additional = do
+ results <-
+ sequence
+ [ approxEq "normalPdf 0 1 0" 0.3989422804 (normalPdf 0 1 0) 1e-9
+ , approxEq "normalPdf 0 1 1" 0.2419707245 (normalPdf 0 1 1) 1e-9
+ , approxEq "expPdf 2 0.5" (2 * exp (-1)) (expPdf 2 0.5) 1e-9
+ , approxEq "poissonPdf 3 2" (4.5 * exp (-3)) (poissonPdf 3 2) 1e-9
+ , approxEq "poissonPdf 1 0" (exp (-1)) (poissonPdf 1 0) 1e-9
+ , -- Six standard errors of the sampling distribution of the mean of 5000 draws.
+ approxEq "normal 3 2 sample mean" 3 (fst normalSampleMoments) (6 * 2 / sqrt 5000)
+ , -- Six standard deviations of the sampling distribution of the variance.
+ approxEq "normal 3 2 sample variance" 4 (snd normalSampleMoments) (6 * sqrt (2 * 4 * 4 / 4999))
+ , assertBool "mh samples stay in [0,1]" =<< smokeMhWithinUnitInterval
+ , assertBool "mhirreducible samples stay in [0,1]" =<< smokeMhIrreducibleWithinUnitInterval
+ , assertBool "wis samples stay in [0,1]" =<< smokeWisWithinUnitInterval
+ , (\n -> approxEq "wis 4 resamples from all four samples" 4 n 0.5) =<< wisDistinctResamples
+ , approxEq "memoize returns f n at n for n in 0..40 (mismatches)" 0 memoizeMismatches 0.5
+ , approxEq "gp with cov 0 0 = 1 has sd 1 at 0" 1 gpStandardDeviationAtZero 0.15
+ , (\d -> approxEq "gp constant covariance 1e8: gap within six nugget standard deviations" 0 d (6*sqrt (2*2**(-33)))) =<< gpConstantKernelGap 1e8
+ , (\d -> approxEq "gp constant covariance 1e20: gap within six nugget standard deviations" 0 d (6*sqrt (2*2**(-33)))) =<< gpConstantKernelGap 1e20
+ , (\d -> approxEq "gp with an RBF kernel scaled by 1e20 stays finite at 31 points" 0 d 0.5) =<< gpScaledKernelFinite 1e20
+ , (\d -> approxEq "gp rank-two covariance 1e16: line residual within six nugget standard deviations" 0 d (6*sqrt (92*2**(-33)))) =<< gpRankTwoLineResidual 1e16
+ , (\d -> approxEq "gp rank-two covariance 1e100: line residual within six nugget standard deviations" 0 d (6*sqrt (92*2**(-33)))) =<< gpRankTwoLineResidual 1e100
+ , (\d -> approxEq "gp with covariance 1e100 cos (x - y) stays finite at 0..5" 0 d 0.5) =<< gpCosineKernelFinite 1e100
+ , (\d -> approxEq "gp scale equivariance for four kernels at 1e-300 to 1e300" 0 d 1e-6) =<< gpScaleEquivarianceError
+ , (\d -> assertBool "gp admits the smallest positive first-point variance" (d > 0 && not (isNaN d || isInfinite d))) =<< gpSmallestVarianceSample
+ , approxEq "ibp 1 first row empty with probability exp (-1)" (exp (-1)) ibpFirstRowEmptyFraction 0.05
+ , assertBool "gp samples stay finite" =<< smokeGpFinite
+ ]
+ finish (additional ++ results)
diff --git a/test/Api/Distributions.hs b/test/Api/Distributions.hs
new file mode 100644
--- /dev/null
+++ b/test/Api/Distributions.hs
@@ -0,0 +1,316 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Api.Distributions where
+
+import Control.Exception (ErrorCall, SomeException, evaluate, try)
+import Data.List (isInfixOf, isPrefixOf, nub, sort, tails)
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import Numeric.Log (Log(..), ln)
+import Numeric (expm1, log1p)
+import System.Timeout (timeout)
+import LazyPPL.Distributions
+import qualified LazyPPL.Distributions.IBP as IBP
+import System.Environment (getArgs, getExecutablePath)
+import System.Process.Typed (proc, readProcessStdout)
+import qualified Data.ByteString.Lazy.Char8 as BL
+import System.Random (RandomGen(..), SplitGen(..), mkStdGen, setStdGen)
+import Control.Monad (replicateM)
+
+import Api.Support
+
+-- 5000 draws of normal 3 2 on a fixed seed, so the sampler is checked, not just
+-- the density.
+normalSampleMoments :: (Double, Double)
+normalSampleMoments =
+ let draws = runProb (replicateM 5000 (normal 3 2)) (randomTree (mkStdGen 7))
+ n = fromIntegral (length draws) :: Double
+ mean = sum draws / n
+ variance = sum [(x - mean) ^ (2 :: Int) | x <- draws] / (n - 1)
+ in (mean, variance)
+
+-- Draws on the simplex. With every label 1/4, normalised Gamma draws underflow
+-- to 0 for small concentrations; a draw must still be a point of the simplex.
+dirichletSamplingChecks :: IO [Bool]
+dirichletSamplingChecks = do
+ let quarter = let t = Tree 0.25 (repeat t) in t
+ sparse = runProb (dirichlet [0.001, 0.001]) quarter
+ single = runProb (dirichlet [0.001]) quarter
+ onSimplex xs = all (\x -> x >= 0 && not (isNaN x || isInfinite x)) xs
+ && abs (sum xs - 1) <= fromIntegral (length xs) * 2.220446049250313e-16
+ points <- sequence
+ [ assertBool ("dirichlet [0.001,0.001] at labels 1/4 is on the simplex, got " ++ show sparse) (onSimplex sparse)
+ , assertBool ("dirichlet [0.001] is exactly [1], got " ++ show single) (single == [1])
+ ]
+ moments <- concat <$> mapM ensemble [[0.5, 1.5, 2], [2, 1.5, 0.5], [0.05, 0.05, 0.1], [0.001, 0.001, 0.002]]
+ pure (points ++ moments)
+ where
+ -- 20000 seeded draws. The means, the variances and the variance of x0 + x1,
+ -- whose law is Beta(a0 + a1, a2), lie within six standard errors of their
+ -- exact values. For a variable in [0,1], E[X^2] bounds n times the variance
+ -- of its sample variance.
+ ensemble alphas = do
+ let n = 20000 :: Int
+ draws = runProb (replicateM n (dirichlet alphas)) (randomTree (mkStdGen 11))
+ total = sum alphas
+ mean xs = sum xs / fromIntegral n
+ variance xs = let m = mean xs in sum [(x - m) ^ (2 :: Int) | x <- xs] / fromIntegral (n - 1)
+ betaVariance a b = a * b / ((a + b) ^ (2 :: Int) * (a + b + 1))
+ coordinate i = map (!! i) draws
+ pairSum = map (\d -> d !! 0 + d !! 1) draws
+ varianceError xs = 6 * sqrt (mean (map (^ (2 :: Int)) xs) / fromIntegral n)
+ name = "dirichlet " ++ show alphas
+ sequence $
+ [ approxEq (name ++ " mean of x" ++ show i) (a / total) (mean (coordinate i))
+ (6 * sqrt (betaVariance a (total - a) / fromIntegral n))
+ | (i, a) <- zip [0 :: Int ..] alphas ] ++
+ [ approxEq (name ++ " variance of x" ++ show i) (betaVariance a (total - a)) (variance (coordinate i))
+ (varianceError (coordinate i))
+ | (i, a) <- zip [0 :: Int ..] alphas ] ++
+ [ approxEq (name ++ " variance of x0 + x1") (betaVariance (alphas !! 0 + alphas !! 1) (alphas !! 2))
+ (variance pairSum) (varianceError pairSum) ]
+
+-- A distribution function on [0,1], extended by 0 below and 1 above.
+onUnit :: (Double -> Double) -> Double -> Double
+onUnit f x = f (max 0 (min 1 x))
+
+-- Beta(1,b) has the distribution function 1 - (1-x)^b and the quantile
+-- 1 - (1-u)^(1/b); Beta(a,1) has x^a and u^(1/a).
+beta1Cdf, betaA1Cdf :: Double -> Double -> Double
+beta1Cdf b = onUnit (\x -> - expm1 (b * log1p (- x)))
+betaA1Cdf a = onUnit (\x -> exp (a * log x))
+
+-- Laws of the Beta and Dirichlet samplers. Each check draws n = 10000 values on
+-- one seeded tree and computes ksDistance from an exact distribution function F.
+-- For n independent draws from F, the Dvoretzky-Kiefer-Wolfowitz inequality with
+-- Massart's constant gives P(sup |F_n - F| > e) <= 2 exp(-2 n e^2): at
+-- e = 0.0269 below 1e-6, and for draws from a law whose distribution function
+-- differs from F by at least 0.0269 + 0.0195 = 0.0464 at a point, a distance
+-- above e with probability at least 0.999. These figures hold for ideal
+-- independent draws; the draws here come from one fixed seed of a pseudorandom
+-- generator. ksDistance compares each draw with F at the draw and at the double
+-- before it, so a difference between two laws inside that interval is not
+-- measured.
+lawChecks :: IO [Bool]
+lawChecks = do
+ let n = 10000 :: Int
+ threshold = sqrt (log (2 / 1e-6) / (2 * fromIntegral n))
+ law name seed cdf p = do
+ let d = ksDistance cdf (runProb (replicateM n p) (randomTree (mkStdGen seed)))
+ assertBool (name ++ ": distance " ++ show d ++ " from the exact distribution function (threshold " ++ show threshold ++ ")")
+ (d <= threshold)
+ coordinate i alphas = (!! i) <$> dirichlet alphas
+ sequence
+ [ law "beta 1 0.001" 21 (beta1Cdf 0.001) (beta 1 0.001)
+ , law "beta 1 0.1" 22 (beta1Cdf 0.1) (beta 1 0.1)
+ , law "beta 0.001 1" 23 (betaA1Cdf 0.001) (beta 0.001 1)
+ , law "beta 0.5 0.5" 24 (onUnit (\x -> 2 / pi * asin (sqrt x))) (beta 0.5 0.5)
+ , law "beta 2 3" 25 (onUnit (\x -> x * x * (6 - 8 * x + 3 * x * x))) (beta 2 3)
+ -- The coordinate of concentration 1 next to 0.001 is Beta(1, 0.001), in either order.
+ , law "dirichlet [1,0.001] x0" 26 (beta1Cdf 0.001) (coordinate 0 [1, 0.001])
+ , law "dirichlet [0.001,1] x1" 27 (beta1Cdf 0.001) (coordinate 1 [0.001, 1])
+ -- x_i is Beta(a_i, a - a_i) with a the total; Beta(2,2) has 3x^2 - 2x^3.
+ , law "dirichlet [0.5,1.5,2] x2" 28 (onUnit (\x -> x * x * (3 - 2 * x))) (coordinate 2 [0.5, 1.5, 2])
+ , law "dirichlet [1,0.01,0.01] x0" 29 (beta1Cdf 0.02) (coordinate 0 [1, 0.01, 0.01])
+ ]
+
+-- The documented domains (LazyPPL.Distributions): dirichlet takes concentrations
+-- of at least 0.001 each with a sum of at most 2000; beta takes the same
+-- parameters, and Beta(a,1) and Beta(1,b) for every finite positive a and b.
+dirichletInDomain, betaInDomain :: Double -> Double -> Bool
+dirichletInDomain a b = a >= 0.001 && b >= 0.001 && a + b <= 2000
+betaInDomain a b = dirichletInDomain a b || (a == 1 && positive b) || (b == 1 && positive a)
+ where positive c = c > 0 && not (isInfinite c)
+
+-- The Beta sampler on constant trees against test/reference/beta-quantiles.tsv.
+-- Inside the documented domain of each, beta a b and the two coordinates of
+-- dirichlet [a,b] at a supported label u lie in the row's intervals: the doubles
+-- whose distribution function differs from u by at most 1e-9 min(u, 1 - u),
+-- joined with the two doubles next to the exact quantile. A row marked reject
+-- has no intervals and checks both samplers' errors instead. Outside the shape
+-- domain or below 2^-53 for general shapes, each raises an error. With
+-- insideOnly, only the rows inside beta's shape domain are read; the
+-- samplers of an earlier version, which have no domain, can take minutes
+-- outside it.
+betaQuantileChecks :: Bool -> IO [Bool]
+betaQuantileChecks insideOnly = do
+ rows <- referenceTable "beta-quantiles.tsv"
+ results <- mapM check [r | r@(a : b : _) <- rows, not insideOnly || betaInDomain (read a) (read b)]
+ tableReport "beta and dirichlet [a,b] in the quantile intervals of beta-quantiles.tsv" results
+ where
+ check [a', b', u', "reject"] = do
+ let (a, b, u) = (read a', read b', read u') :: (Double, Double, Double)
+ tree = constantLabel u
+ unsupportedLabel = u < encodeFloat 1 (-53) && a /= 1 && b /= 1
+ betaSupportedHere = betaInDomain a b && not unsupportedLabel
+ dirichletSupportedHere = dirichletInDomain a b && not unsupportedLabel
+ single <- if betaInDomain a b
+ then raisesBetaLabel (runProb (beta a b) tree)
+ else raisesError (runProb (beta a b) tree)
+ pair <- if dirichletInDomain a b
+ then raisesBetaLabel (sum (runProb (dirichlet [a,b]) tree))
+ else raisesError (sum (runProb (dirichlet [a,b]) tree))
+ pure (not betaSupportedHere && not dirichletSupportedHere && single && pair,
+ "beta and dirichlet must reject " ++ show (a,b,u))
+ check (a' : b' : u' : xlo' : xhi' : ylo' : yhi' : _) = do
+ let (a, b, u, xlo, xhi, ylo, yhi) = (read a', read b', read u', read xlo', read xhi', read ylo', read yhi') :: (Double, Double, Double, Double, Double, Double, Double)
+ tree = constantLabel u
+ name = "beta " ++ show a ++ " " ++ show b ++ " at label " ++ show u
+ inside lo hi v = lo <= v && v <= hi
+ answer v = maybe "an error or no answer in 2 s" show v
+ intervals = "x in [" ++ show xlo ++ "," ++ show xhi ++ "] and 1-x in [" ++ show ylo ++ "," ++ show yhi ++ "]"
+ let unsupportedLabel = u < encodeFloat 1 (-53) && a /= 1 && b /= 1
+ single <- if betaInDomain a b && unsupportedLabel
+ then (\e -> (e, "general beta rejects the custom label below 2^-53")) <$> raisesBetaLabel (runProb (beta a b) tree)
+ else if betaInDomain a b
+ then do
+ x <- within 2000000 (runProb (beta a b) tree)
+ pure (maybe False (inside xlo xhi) x, "beta: " ++ intervals ++ ", got " ++ answer x)
+ else (\e -> (e, "beta is outside its domain and must raise an error")) <$> raisesError (runProb (beta a b) tree)
+ pair <- if dirichletInDomain a b && unsupportedLabel
+ then (\e -> (e, "dirichlet rejects the custom label below 2^-53")) <$> raisesBetaLabel (sum (runProb (dirichlet [a, b]) tree))
+ else if dirichletInDomain a b
+ then do
+ xs <- within 2000000 (let xs = runProb (dirichlet [a, b]) tree in sum xs `seq` xs)
+ pure (case xs of
+ Just [v, w] -> inside xlo xhi v && inside ylo yhi w
+ _ -> False, "dirichlet: " ++ intervals ++ ", got " ++ answer xs)
+ else (\e -> (e, "dirichlet is outside its domain and must raise an error")) <$> raisesError (sum (runProb (dirichlet [a, b]) tree))
+ pure (fst single && fst pair, name ++ ": " ++ snd single ++ "; " ++ snd pair)
+ check row = pure (False, "malformed row " ++ show row)
+
+-- dirichlet as at the constant label u against the exact stick-breaking
+-- coordinates of test/reference/dirichlet-sticks.tsv. Coordinate i is the
+-- remaining stick times the u-quantile of Beta(a_i, a_(i+1) + ... + a_k), and
+-- its interval adds the relative tolerances of the shares that form it.
+dirichletStickChecks :: IO [Bool]
+dirichletStickChecks = do
+ rows <- referenceTable "dirichlet-sticks.tsv"
+ results <- mapM check rows
+ tableReport "dirichlet at the exact stick-breaking coordinates of dirichlet-sticks.tsv" results
+ where
+ check (alphas' : u' : bounds') = do
+ let alphas = read ("[" ++ alphas' ++ "]") :: [Double]
+ u = read u' :: Double
+ bounds = intervals (map read bounds')
+ unsupportedLabel = u < encodeFloat 1 (-53) && any (\(a, rest) -> a /= 1 && foldr (+) 0 rest /= 1)
+ [(a, rest) | (a : rest) <- tails alphas, not (null rest)]
+ if unsupportedLabel
+ then do
+ rejected <- raisesBetaLabel (sum (runProb (dirichlet alphas) (constantLabel u)))
+ pure (rejected, "dirichlet " ++ show alphas ++ " must reject label " ++ show u)
+ else do
+ xs <- within 2000000 (let v = runProb (dirichlet alphas) (constantLabel u) in sum v `seq` v)
+ let ok = maybe False (\v -> length v == length bounds && and (zipWith (\c (lo, hi) -> lo <= c && c <= hi) v bounds)) xs
+ pure (ok, "dirichlet " ++ show alphas ++ " at label " ++ show u ++ ": coordinates in " ++ show bounds
+ ++ ", got " ++ maybe "an error or no answer in 2 s" show xs)
+ check row = pure (False, "malformed row " ++ show row)
+ intervals (lo : hi : rest) = (lo, hi) : intervals rest
+ intervals _ = [] :: [(Double, Double)]
+
+-- For sorted xs and sorted points, how many xs are at most each point.
+atOrBelow :: [Double] -> [Double] -> [Int]
+atOrBelow = go 0
+ where
+ go _ _ [] = []
+ go k xs (p : ps) = let (le, rest) = span (<= p) xs; k' = k + length le in k' : go k' rest ps
+
+-- Laws of the Beta sampler: 10000 seeded draws against the exact distribution
+-- function of test/reference/beta-laws.tsv at its grid of points, for two shapes
+-- without a closed form and for Beta(1,b) and Beta(a,1) at 1e-4 and 1e4, the
+-- sticks of the Dirichlet process and of IBP's newRestaurantS. The largest distance over the grid is at most the Kolmogorov-Smirnov
+-- statistic, so the threshold of lawChecks keeps its false-alarm bound; the
+-- grid's points are close enough that the distribution function rises by less
+-- than 0.005 between neighbours.
+gridLawChecks :: IO [Bool]
+gridLawChecks = do
+ rows <- referenceTable "beta-laws.tsv"
+ let grid = [((read a, read b), (read x, read f)) | [a, b, x, f] <- rows] :: [((Double, Double), (Double, Double))]
+ n = 10000 :: Int
+ threshold = sqrt (log (2 / 1e-6) / (2 * fromIntegral n))
+ shapes = [(2.5, 3.5), (1000, 1000), (1, 1e-4), (1, 1e4), (1e-4, 1), (1e4, 1)]
+ coverage <- assertBool "beta-laws.tsv covers both general shapes and all four nonparametric stick laws"
+ (sort (nub (map fst grid)) == sort shapes)
+ results <- sequence
+ [ do let points = [p | (s, p) <- grid, s == (a, b)]
+ draws = sort (runProb (replicateM n (beta a b)) (randomTree (mkStdGen seed)))
+ d = maximum (zipWith (\k (_, f) -> abs (fromIntegral k / fromIntegral n - f)) (atOrBelow draws (map fst points)) points)
+ result <- try (evaluate d)
+ case result of
+ Right distance -> assertBool ("beta " ++ show a ++ " " ++ show b ++ ": distance " ++ show distance ++ " from the exact distribution function at "
+ ++ show (length points) ++ " points (threshold " ++ show threshold ++ ")") (distance <= threshold)
+ Left err -> assertBool ("beta " ++ show a ++ " " ++ show b ++ ": " ++ takeWhile (/= '\n') (show (err :: ErrorCall))) False
+ | (seed, (a, b)) <- zip [41 ..] (nub (map fst grid)) ]
+ pure (coverage : results)
+
+-- One row of test/reference/log-densities.tsv: the function at the arguments is
+-- within 1e-12 max(1, |exact|) of the exact value, or is the same infinity where
+-- the exact value leaves the double range.
+densityRow :: [String] -> (Bool, String)
+densityRow [function, arguments, exact'] =
+ let expected = read exact' :: Double
+ actual = case (function, words arguments) of
+ ("poissonLogPmf", [l, k]) -> poissonLogPmf (read l) (read k)
+ ("gammaLogPdf", [k, t, x]) -> gammaLogPdf (read k) (read t) (read x)
+ ("betaLogPdf", [a, b, x]) -> betaLogPdf (read a) (read b) (read x)
+ ("binomialLogPmf", [n, p, k]) -> binomialLogPmf (read n) (read p) (read k)
+ ("dirichletLogPdf", [as, xs]) -> dirichletLogPdf (read as) (read xs)
+ _ -> error ("log-densities.tsv: unknown row " ++ function ++ " " ++ arguments)
+ ok | isInfinite expected = actual == expected
+ | otherwise = abs (actual - expected) <= 1e-12 * max 1 (abs expected)
+ in (ok, function ++ " " ++ arguments ++ " is " ++ show expected ++ ", got " ++ show actual)
+densityRow row = (False, "malformed row " ++ show row)
+
+-- The rows of test/reference/log-densities.tsv run in a child process, this
+-- program with the argument "density-rows", which prints one verdict per row.
+-- A computation that loops without allocating cannot be interrupted inside its
+-- own process (poissonLogPmf 1e308 (10^308) once did), so the deadline is kept
+-- from outside: a child that has not exited within 120 s is terminated, and
+-- every row without a verdict fails.
+densityChecks :: IO [Bool]
+densityChecks = do
+ rows <- referenceTable "log-densities.tsv"
+ out <- externally 120000000 ["density-rows"]
+ let reported = [l | l <- maybe [] lines out, take 4 l `elem` ["PASS", "FAIL"]]
+ missing = length rows - length reported
+ mapM_ putStrLn [l | l <- reported, take 4 l == "FAIL"]
+ putStrLn ((if missing == 0 && all ((== "PASS") . take 4) reported then "PASS " else "FAIL ")
+ ++ "log densities of log-densities.tsv: " ++ show (length (filter ((== "PASS") . take 4) reported))
+ ++ " of " ++ show (length rows) ++ " rows"
+ ++ (if missing > 0 then ", " ++ show missing ++ " without a verdict" ++ maybe " (no exit within 120 s)" (const "") out else ""))
+ pure (map ((== "PASS") . take 4) reported ++ replicate missing False)
+
+densityRowsChild :: IO ()
+densityRowsChild = do
+ rows <- referenceTable "log-densities.tsv"
+ mapM_ (\row -> let (ok, detail) = densityRow row in putStrLn ((if ok then "PASS " else "FAIL ") ++ detail)) rows
+
+-- Runs this test program with the given arguments and returns its standard
+-- output, or Nothing when it has not exited within the given microseconds, in
+-- which case it is terminated. typed-process reads the output while the child
+-- runs, so a full pipe cannot stop it.
+externally :: Int -> [String] -> IO (Maybe String)
+externally micros args = do
+ self <- getExecutablePath
+ fmap (BL.unpack . snd) <$> timeout micros (readProcessStdout (proc self args))
+
+-- Log densities and draws at extreme but valid parameters. Each expected value
+-- comes from the closed form or asymptotic expansion named beside it.
+numericChecks :: IO [Bool]
+numericChecks = do
+ let half = let t = Tree 0.5 (repeat t) in t
+ relative name expected actual = approxEq name expected actual (1e-12 * max 1 (abs expected))
+ exactly name expected actual = assertBool (name ++ " is " ++ show expected ++ ", got " ++ show actual) (actual == expected)
+ sequence
+ [ -- z = 1: -log(2 pi)/2 - log 1e200 - 1/2
+ relative "normalLogPdf 0 1e200 1e200" (-461.9359571320138) (normalLogPdf 0 1e200 1e200)
+ -- z = 0: -log(2 pi)/2 + log 1e200
+ , relative "normalLogPdf 0 1e-200 0" 459.5980800656045 (normalLogPdf 0 1e-200 0)
+ , relative "logNormalLogPdf 0 1e-200 1" 459.5980800656045 (logNormalLogPdf 0 1e-200 1)
+ -- z = 2 while x - mu exceeds the largest double: -log(2 pi)/2 - log 1e308 - 2
+ , relative "normalLogPdf -1e308 1e308 1e308" (-712.1151471753708) (normalLogPdf (-1e308) 1e308 1e308)
+ -- the median of a normal is its mean, up to the quantile's absolute accuracy near 0
+ , let v = runProb (normal 0 1.5e308) half
+ in assertBool ("normal 0 1.5e308 at label 1/2 is within 1e-15 sigma of 0, got " ++ show v) (abs v <= 1e-15 * 1.5e308)
+ -- width 2e308: -log 2 - log 1e308
+ , approxEq "uniformBoundedLogPdf -1e308 1e308 0" (-709.8893558227261) (uniformBoundedLogPdf (-1e308) 1e308 0) 1e-9
+ , exactly "uniformbounded -1e308 1e308 at label 1/2" 0 (runProb (uniformbounded (-1e308) 1e308) half)
+ ]
diff --git a/test/Api/Domains.hs b/test/Api/Domains.hs
new file mode 100644
--- /dev/null
+++ b/test/Api/Domains.hs
@@ -0,0 +1,352 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Api.Domains where
+
+import Control.Exception (ErrorCall, SomeException, evaluate, try)
+import Data.List (foldl', isInfixOf, isPrefixOf, nub, sort, tails)
+import Data.Word (Word64)
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import Control.Monad.Trans.Writer (runWriterT)
+import Data.Monoid (Product(..), getProduct)
+import Numeric.Log (Log(..), ln)
+import qualified Numeric.Sum as Sum
+import System.Timeout (timeout)
+import LazyPPL.Distributions
+import qualified LazyPPL.Distributions.IBP as IBP
+import qualified LazyPPL.Distributions.DirichletP as DP
+import System.Random (RandomGen(..), SplitGen(..), mkStdGen, setStdGen)
+import Control.Monad (replicateM)
+import Test.QuickCheck (NonEmptyList(..), NonNegative(..), Positive(..), Property, ioProperty, isSuccess, quickCheckResult)
+
+import Api.Support
+
+-- An impossible event has mass 0, so its log is negative infinity.
+densityBoundaryChecks :: IO [Bool]
+densityBoundaryChecks = sequence
+ [ assertBool name (isInfinite actual && actual < 0)
+ | (name, actual) <-
+ [ ("Bernoulli impossible success", bernoulliLogPmf 0 True)
+ , ("Bernoulli impossible failure", bernoulliLogPmf 1 False)
+ , ("categorical zero mass", categoricalLogPmf [1,0] 1)
+ , ("categorical index outside support", categoricalLogPmf [1] 1)
+ , ("Poisson positive count at rate zero", poissonLogPmf 0 1)
+ , ("Dirichlet outside simplex", dirichletLogPdf [1,1,1] [0.2,0.2,0.2])
+ , ("Dirichlet empty simplex", dirichletLogPdf [] [])
+ , ("exponential log density outside support", exponentialLogPdf 1 (-1))
+ ]
+ ] >>= \results -> (results ++) <$> sequence
+ [ approxEq "Poisson zero rate and zero count" 0 (poissonLogPmf 0 0) 1e-15
+ , approxEq "Poisson density zero rate and zero count" 1 (poissonPdf 0 0) 1e-15
+ , do
+ result <- try (evaluate (poissonPdf 1 (-1))) :: IO (Either ErrorCall Double)
+ case result of
+ Left err -> putStrLn ("FAIL Poisson negative count: " ++ show err) >> pure False
+ Right value -> approxEq "Poisson negative count" 0 value 1e-15
+ , approxEq "exponential density outside support" 0 (expPdf 1 (-1)) 1e-15
+ , approxEq "Dirichlet(1,1,1) simplex density" (log 2) (dirichletLogPdf [1,1,1] [0.25,0.25,0.5]) 1e-12
+ , approxEq "binomial certain failure" 0 (binomialLogPmf 5 0 0) 1e-15
+ , approxEq "binomial certain success" 0 (binomialLogPmf 5 1 5) 1e-15
+ , let Meas action = score 0
+ (_, weight) = runProb (runWriterT action) (randomTree (mkStdGen 1))
+ in assertBool "score zero stays impossible" (ln (getProduct weight) == -1/0)
+ , and <$> mapM (\x -> do
+ let Meas action = score x
+ (_, weight) = runProb (runWriterT action) (randomTree (mkStdGen 1))
+ result <- try (evaluate (ln (getProduct weight))) :: IO (Either ErrorCall Double)
+ assertBool "invalid score is rejected" (either (const True) (const False) result)) [-1, 0/0, 1/0]
+ , isSuccess <$> quickCheckResult dirichletNormalizedInput
+ ]
+
+dirichletNormalizedInput :: NonEmptyList (Positive Int) -> Bool
+dirichletNormalizedInput (NonEmpty positive) =
+ let values = map (fromIntegral . getPositive) positive :: [Double]
+ xs = map (/ sum values) values
+ density = dirichletLogPdf (replicate (length xs) 1) xs
+ in not (isNaN density || isInfinite density)
+
+-- The Dirichlet process scores atoms on an infinite stick-breaking probability
+-- stream, so `categoricalLogPmf` must inspect only the prefix up to the selected
+-- element and return on an infinite stream.
+lazyCategoricalChecks :: IO [Bool]
+lazyCategoricalChecks = do
+ let stream = 0 : 1 : repeat 0
+ selected <- evaluate (categoricalLogPmf stream 1)
+ zeroMass <- evaluate (categoricalLogPmf stream 0)
+ invalid <- try (evaluate (categoricalLogPmf (1.5 : repeat 0) 0)) :: IO (Either ErrorCall Double)
+ sequence
+ [ assertBool "categoricalLogPmf scores an infinite probability stream" (selected == log 1)
+ , assertBool "categoricalLogPmf gives negative infinity for zero mass on an infinite stream"
+ (isInfinite zeroMass && zeroMass < 0)
+ , assertBool "categoricalLogPmf rejects an invalid inspected probability"
+ (either (const True) (const False) invalid)
+ ]
+
+-- A generator that returns one word forever, so every label of the tree is the
+-- label of that word.
+newtype ConstWord = ConstWord Word64
+
+instance RandomGen ConstWord where
+ genWord64 g@(ConstWord w) = (w, g)
+
+instance SplitGen ConstWord where
+ splitGen g = (g, g)
+
+categoricalBoundaryChecks :: IO [Bool]
+categoricalBoundaryChecks = do
+ let e = encodeFloat 1 (-54)
+ weights = [0.5, e, e, e, e, 0.5 - 4 * e]
+ tree = randomTree (ConstWord 0)
+ result <- try (timeout 500000 (evaluate (runProb (categorical weights) tree))) :: IO (Either SomeException (Maybe Int))
+ sequence
+ [ assertBool ("categorical selects the final positive weight at the maximum label, got " ++ show result)
+ (case result of Right (Just 5) -> True; _ -> False)
+ ]
+
+categoricalFiniteProperty :: NonEmptyList Word64 -> Word64 -> Property
+categoricalFiniteProperty (NonEmpty values) word = ioProperty $ do
+ let masses = 1 : map (fromIntegral . (`mod` 1024)) values
+ weights = map (/ sum masses) masses
+ labels = [0, 1 - encodeFloat 1 (-53), runProb uniform (randomTree (ConstWord word))]
+ valid (Just i) = i >= 0 && i < length weights && weights !! i > 0
+ valid Nothing = False
+ draws <- mapM (\label -> within 500000 (runProb (categorical weights) (constantLabel label))) labels
+ pure (all valid draws)
+
+categoricalInfiniteProperty :: NonEmptyList Word64 -> NonNegative Int -> Word64 -> Property
+categoricalInfiniteProperty (NonEmpty values) (NonNegative prefix) word = ioProperty $ do
+ let splitMass n value =
+ let mass = encodeFloat 1 (-n)
+ fraction = fromIntegral (value `mod` 17) / 16
+ in [mass * fraction, mass * (1 - fraction)]
+ weights = replicate (prefix `mod` 100) 0 ++ concat (zipWith splitMass [1..] (cycle values))
+ labels = [0, 1 - encodeFloat 1 (-53), runProb uniform (randomTree (ConstWord word))]
+ valid (Just i) = i >= 0 && weights !! i > 0
+ valid Nothing = False
+ draws <- mapM (\label -> within 500000 (runProb (categorical weights) (constantLabel label))) labels
+ pure (all valid draws)
+
+categoricalPropertyChecks :: IO [Bool]
+categoricalPropertyChecks = do
+ finite <- quickCheckResult categoricalFiniteProperty
+ infinite <- quickCheckResult categoricalInfiniteProperty
+ shortDraw <- try (evaluate (runProb (categorical [0.5, 0.3]) (constantLabel 0.9))) :: IO (Either ErrorCall Int)
+ quarterShort <- try (evaluate (runProb (categorical [0.5, 0.25]) (constantLabel 0.9999999999999999))) :: IO (Either ErrorCall Int)
+ tailDraw <- try (evaluate (runProb (categorical [0.2, 0, 0]) (constantLabel 0.9))) :: IO (Either ErrorCall Int)
+ let nearOne = [0.5, 0.5 - encodeFloat 1 (-40)]
+ maxLabel = constantLabel (1 - encodeFloat 1 (-53))
+ oneUlpDraw <- try (evaluate (runProb (categorical [1 - encodeFloat 1 (-53)]) maxLabel)) :: IO (Either ErrorCall Int)
+ paddedDraw <- try (evaluate (runProb (categorical (nearOne ++ replicate 8190 0)) maxLabel)) :: IO (Either ErrorCall Int)
+ tinyDraw <- try (evaluate (runProb (categorical (nearOne ++ replicate 8190 (encodeFloat 1 (-1074)))) maxLabel)) :: IO (Either ErrorCall Int)
+ earlyDraw <- within 500000 (runProb (categorical [0.5, 0.3]) (constantLabel 0.4))
+ lazyDraw <- within 500000 (runProb (categorical (0.8 : error "categorical forced the unchosen tail")) (constantLabel 0.4))
+ roundedDraw <- within 500000 (runProb (categorical [1/590, 459/590, 130/590]) (constantLabel (1 - encodeFloat 1 (-53))))
+ decimalDraw <- within 500000 (runProb (categorical [0.1, 0.2, 0.7]) (constantLabel (1 - encodeFloat 1 (-53))))
+ decimalFallback <- within 500000 (runProb (categorical [0.50000000000000005, 0.250000000000000025, 0.249999999999999925]) maxLabel)
+ let equalWeights = replicate 49 (1/49) :: [Double]
+ equalTotal = Sum.kbn (foldl' Sum.add Sum.zero equalWeights)
+ equalDraw <- within 500000 (runProb (categorical equalWeights) maxLabel)
+ negativeZeroDraw <- within 500000 (runProb (categorical ((-0.0) : replicate 49 (1/49))) maxLabel)
+ allZero <- rejects "categorical [0,0]" (runProb (categorical [0, 0]) (constantLabel 0.9))
+ sequence
+ [ assertBool "categorical chooses positive finite weights" (isSuccess finite)
+ , assertBool "categorical chooses positive weights from an infinite stream" (isSuccess infinite)
+ , assertBool ("categorical rejects missing mass in [0.5,0.3], got " ++ show shortDraw)
+ (case shortDraw of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool "categorical rejects [0.5,0.25] at the largest label"
+ (case quarterShort of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool ("categorical rejects missing mass before trailing zeros, got " ++ show tailDraw)
+ (case tailDraw of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool "categorical rejects a singleton one ulp below one"
+ (case oneUlpDraw of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool "categorical rejects a clear deficit despite many trailing zeros"
+ (case paddedDraw of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool "categorical rejects a clear deficit despite many tiny weights"
+ (case tinyDraw of Left err -> "probabilities do not sum to 1" `isInfixOf` show err; _ -> False)
+ , assertBool "categorical stops before a later deficit" (earlyDraw == Just 0)
+ , assertBool "categorical does not force an unchosen tail" (lazyDraw == Just 0)
+ , assertBool "categorical draws from a rounded normalized vector" (roundedDraw == Just 2)
+ , assertBool "categorical accepts rounded decimal probabilities" (decimalDraw == Just 2)
+ , assertBool "categorical accepts rounded decimal probabilities at exhaustion" (decimalFallback == Just 2)
+ , assertBool ("categorical 49-weight compensated sum is 0.9999999999999999, got " ++ show equalTotal)
+ (equalTotal == 0.9999999999999999)
+ , assertBool "categorical selects index 48 for 49 rounded equal probabilities at 0.9999999999999999" (equalDraw == Just 48)
+ , assertBool "categorical accepts negative zero before rounded probabilities" (negativeZeroDraw == Just 49)
+ , pure allZero
+ ]
+
+dirichletStickRoundingChecks :: IO [Bool]
+dirichletStickRoundingChecks = do
+ let e = encodeFloat 1 (-54)
+ tree = Tree 0.5 (branch (0.5 : replicate 8 e ++ repeat 0.5) : repeat (constantLabel 0.5))
+ branch (u:us) = Tree 0.5 (stickTree u : branch us : repeat (constantLabel 0.5))
+ branch [] = constantLabel 0.5
+ stickTree u = Tree 0.5 (constantLabel u : repeat (constantLabel 0.5))
+ restaurant = runProb (DP.newRestaurant 1) tree
+ draw u = runProb (DP.newCustomer restaurant) (constantLabel u)
+ atBoundary <- within 500000 (draw (0.75 + 2 * e))
+ insideNext <- within 500000 (draw 0.8)
+ beforeBoundary <- within 500000 (draw 0.75)
+ sequence [assertBool ("Dirichlet process retains the small sticks before the boundary: " ++ show beforeBoundary ++ ", " ++ show atBoundary ++ ", " ++ show insideNext) (atBoundary == insideNext && atBoundary /= Nothing && atBoundary /= beforeBoundary && beforeBoundary /= Nothing)]
+
+poissonBoundaryChecks :: IO [Bool]
+poissonBoundaryChecks = do
+ result <- within 500000 (runProb (poisson 1) (constantLabel (1 - encodeFloat 1 (-53))))
+ low <- within 500000 (runProb (poisson 36.5) (constantLabel (encodeFloat 1 (-53))))
+ sequence
+ [ assertBool ("Poisson(1) at the largest label gives 17, got " ++ show result) (result == Just 17)
+ , assertBool ("Poisson(36.5) at the smallest label gives 0, got " ++ show low) (low == Just 0)
+ ]
+
+-- Words 0 and 2^64-1 are the two ends of the generator's range. Every label of
+-- the tree lies strictly between 0 and 1, so each inverse transform returns a
+-- value in its support.
+endpointChecks :: IO [Bool]
+endpointChecks = concat <$> mapM check [("word 0", 0), ("word 2^64-1", maxBound)]
+ where
+ check (name, word) = do
+ let tree = randomTree (ConstWord word)
+ draw p = within 500000 (runProb p tree)
+ label = runProb uniform tree
+ finite x = not (isNaN x || isInfinite x)
+ discrete <- draw (uniformdiscrete 1)
+ sure <- draw (bernoulli 1)
+ never <- draw (bernoulli 0)
+ single <- draw (categorical [1])
+ count <- draw (poisson 0)
+ waiting <- draw (exponential 1)
+ gaussian <- draw (normal 0 1)
+ sequence
+ [ assertBool (name ++ ": label " ++ show label ++ " lies in (0,1)") (label > 0 && label < 1)
+ , assertBool (name ++ ": uniformdiscrete 1 gives 0, got " ++ show discrete) (discrete == Just 0)
+ , assertBool (name ++ ": bernoulli 1 gives True, got " ++ show sure) (sure == Just True)
+ , assertBool (name ++ ": bernoulli 0 gives False, got " ++ show never) (never == Just False)
+ , assertBool (name ++ ": categorical [1] gives 0, got " ++ show single) (single == Just 0)
+ , assertBool (name ++ ": poisson 0 gives 0, got " ++ show count) (count == Just 0)
+ , assertBool (name ++ ": exponential 1 is finite, got " ++ show waiting) (maybe False finite waiting)
+ , assertBool (name ++ ": normal 0 1 is finite, got " ++ show gaussian) (maybe False finite gaussian)
+ ]
+
+-- A log weight is finite, or negative infinity for weight 0. NaN and positive
+-- infinity are not weights, and a sampler must not run on them.
+logWeightChecks :: IO [Bool]
+logWeightChecks = do
+ let firstLogWeight model = do
+ setStdGen (mkStdGen 1)
+ draws <- mh 1 model
+ evaluate (ln (getProduct (snd (head draws))))
+ scoredBy scorer = do x <- sample uniform; scorer; pure x
+ invalid = [("NaN", Exp (0/0)), ("+Infinity", Exp (1/0))]
+ single <- sequence
+ [ rejectsIO ("mh rejects scoreLog " ++ name) (firstLogWeight (scoredBy (scoreLog w))) | (name, w) <- invalid ]
+ products <- sequence
+ [ rejectsIO ("mh rejects scoreProductLog " ++ name) (firstLogWeight (scoredBy (scoreProductLog (Product w))))
+ | (name, w) <- invalid ]
+ overflow <- rejectsIO "mh rejects a total log weight that overflows to +Infinity"
+ (firstLogWeight (scoredBy (scoreLog (Exp 1e308) >> scoreLog (Exp 1e308))))
+ large <- firstLogWeight (scoredBy (scoreLog (Exp 1000)))
+ zero <- firstLogWeight (scoredBy (scoreLog (Exp (-1/0))))
+ valid <- sequence
+ [ assertBool ("mh keeps a log weight of 1000, got " ++ show large) (large == 1000)
+ , assertBool ("mh keeps weight 0 as log weight -Infinity, got " ++ show zero) (isInfinite zero && zero < 0)
+ ]
+ pure (single ++ products ++ [overflow] ++ valid)
+
+-- The chances given to mh and mhirreducible are probabilities, and the
+-- irreducible kernel needs a positive chance q of an all-sites change.
+chanceChecks :: IO [Bool]
+chanceChecks = do
+ let firstValue sampler = do
+ setStdGen (mkStdGen 1)
+ draws <- sampler (sample uniform)
+ evaluate (fst (head draws))
+ invalid <- sequence
+ [ rejectsIO (name ++ " is rejected") (firstValue sampler)
+ | (name, sampler) <-
+ [ ("mhirreducible 0 0", mhirreducible 0 0)
+ , ("mhirreducible 0.5 NaN", mhirreducible 0.5 (0/0))
+ , ("mhirreducible 0.5 1.5", mhirreducible 0.5 1.5)
+ , ("mh NaN", mh (0/0))
+ , ("mh -0.5", mh (-0.5))
+ , ("mh 1.5", mh 1.5) ] ]
+ setStdGen (mkStdGen 1)
+ moving <- take 16 <$> mhirreducible 0 0.5 (sample uniform)
+ valid <- assertBool "mhirreducible 0 0.5 moves through its all-sites changes"
+ (length (nub (map fst moving)) > 1)
+ pure (invalid ++ [valid])
+
+betaLabelChecks :: IO [Bool]
+betaLabelChecks = do
+ let least = encodeFloat 1 (-1074)
+ at u p = runProb p (constantLabel u)
+ rejected name value = raisesBetaLabel value >>= assertBool (name ++ " raises BetaLabelError")
+ sequence
+ [ rejected "beta 2 2 at least subnormal" (at least (beta 2 2))
+ , rejected "beta 1000 1000 at least subnormal" (at least (beta 1000 1000))
+ , rejected "beta 2 2 at 1e-320" (at 1e-320 (beta 2 2))
+ , rejected "dirichlet [2,2] at 1e-320" (sum (at 1e-320 (dirichlet [2,2])))
+ , assertBool "beta 1 2 rounds the least-label quantile to zero" (at least (beta 1 2) == 0)
+ , approxEq "beta 2 1 at the least label" (encodeFloat 1 (-537)) (at least (beta 2 1)) (encodeFloat 1 (-537) * 1e-12)
+ ]
+
+-- The Dirichlet process and IBP's newRestaurantS draw their sticks from
+-- beta 1 alpha and beta alpha 1. They sample at every finite positive
+-- concentration, 1e-4 and 1e4 among them, and raise an error that names the
+-- function and alpha for any other.
+concentrationChecks :: IO [Bool]
+concentrationChecks = do
+ let tree = randomTree (mkStdGen 7)
+ samples name p = do
+ r <- within 2000000 (let xs = runProb p tree in length (show xs) `seq` xs)
+ assertBool (name ++ " samples") (maybe False (const True) r)
+ namesItself function name x = do
+ result <- try (timeout 2000000 (evaluate x))
+ let message = either (\err -> takeWhile (/= '\n') (show (err :: ErrorCall))) (const "") result
+ assertBool (name ++ " raises an error naming " ++ function ++ " and alpha"
+ ++ either (const (": " ++ message)) (const " (it returned a value or no answer)") result)
+ (function `isPrefixOf` message && "alpha" `isInfixOf` message)
+ sequence $
+ concat [ [ samples ("DirichletP.newRestaurant " ++ show alpha ++ " with 20 customers")
+ (DP.newRestaurant alpha >>= \r -> replicateM 20 (DP.newCustomer r))
+ , samples ("dp " ++ show alpha ++ " (normal 0 1), 20 draws")
+ (DP.dp alpha (normal 0 1) >>= \d -> replicateM 20 d)
+ , samples ("IBP.newRestaurantS " ++ show alpha ++ ", the first 10 dishes of 3 customers")
+ (IBP.newRestaurantS alpha >>= \r -> replicateM 3 (take 10 <$> IBP.newCustomerS r)) ]
+ | alpha <- [1e-4, 1e4] ] ++
+ [ namesItself "newRestaurant" "DirichletP.newRestaurant 0" (runProb (DP.newRestaurant 0 >>= DP.newCustomer) tree)
+ , namesItself "dp" "dp (-1) (normal 0 1)" (runProb (DP.dp (-1) (normal 0 1) >>= id) tree)
+ , namesItself "newRestaurantS" "IBP.newRestaurantS NaN" (runProb (IBP.newRestaurantS (0 / 0) >>= IBP.newCustomerS) tree)
+ , namesItself "dp" "dp Infinity (normal 0 1)" (runProb (DP.dp (1 / 0) (normal 0 1) >>= id) tree) ]
+
+-- Each distribution checks its parameters, so an invalid parameter raises an
+-- error instead of silently changing the law; samplers and densities agree.
+parameterChecks :: IO [Bool]
+parameterChecks = do
+ let tree = randomTree (mkStdGen 1)
+ sampled :: Prob a -> a
+ sampled p = runProb p tree
+ sequence
+ [ rejects "bernoulli 2" (sampled (bernoulli 2))
+ , rejects "bernoulli -0.5" (sampled (bernoulli (-0.5)))
+ , rejects "exponential -1" (sampled (exponential (-1)))
+ , rejects "exponential 0" (sampled (exponential 0))
+ , rejects "gamma 1 -1" (sampled (gamma 1 (-1)))
+ , rejects "gamma -1 1" (sampled (gamma (-1) 1))
+ , rejects "beta 0 1" (sampled (beta 0 1))
+ -- Outside the supported domain of beta and dirichlet: large shapes, and
+ -- concentrations whose sum overflows.
+ , rejects "beta 1e16 1e16" (sampled (beta 1e16 1e16))
+ , rejects "dirichlet [1e308,1e308,1e308]" (sampled (dirichlet [1e308, 1e308, 1e308]))
+ , rejects "normal 0 -1" (sampled (normal 0 (-1)))
+ , rejects "normal 0 0" (sampled (normal 0 0))
+ , rejects "poisson -1" (sampled (poisson (-1)))
+ , rejects "uniformdiscrete 0" (sampled (uniformdiscrete 0))
+ , rejects "uniformbounded 1 0" (sampled (uniformbounded 1 0))
+ , rejects "categorical [-1,2]" (sampled (categorical [-1, 2]))
+ , rejects "categorical [2,-1]" (sampled (categorical [2, -1]))
+ , rejects "normalLogPdf 0 -1 0" (normalLogPdf 0 (-1) 0)
+ , rejects "logNormalLogPdf 0 0 1" (logNormalLogPdf 0 0 1)
+ , rejects "gammaLogPdf -1 1 1" (gammaLogPdf (-1) 1 1)
+ , rejects "betaLogPdf 0 1 0.5" (betaLogPdf 0 1 0.5)
+ , rejects "betaLogPdf 1e308 1e308 0.5" (betaLogPdf 1e308 1e308 0.5)
+ , rejects "dirichletLogPdf [1e308,1e308] [0.5,0.5]" (dirichletLogPdf [1e308, 1e308] [0.5, 0.5])
+ , rejects "uniformBoundedLogPdf 1 0 0.5" (uniformBoundedLogPdf 1 0 0.5)
+ ]
diff --git a/test/Api/Inference.hs b/test/Api/Inference.hs
new file mode 100644
--- /dev/null
+++ b/test/Api/Inference.hs
@@ -0,0 +1,119 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Api.Inference where
+
+import Control.Exception (ErrorCall, SomeException, evaluate, try)
+import Data.List (isInfixOf, isPrefixOf, nub, sort, tails)
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import Data.Monoid (Product(..), getProduct)
+import Numeric.Log (Log(..), ln)
+import LazyPPL.Distributions
+import System.Random (RandomGen(..), SplitGen(..), mkStdGen, setStdGen)
+import Control.Monad (replicateM)
+
+import Api.Support
+
+smokeMhWithinUnitInterval :: IO Bool
+smokeMhWithinUnitInterval = do
+ draws <- take 25 <$> mh 0.5 (sample uniform)
+ pure $ all (\(x, _) -> x >= 0 && x <= 1) draws
+
+smokeMhIrreducibleWithinUnitInterval :: IO Bool
+smokeMhIrreducibleWithinUnitInterval = do
+ draws <- take 25 <$> mhirreducible 0.5 0.1 (sample uniform)
+ pure $ all (\(x, _) -> x >= 0 && x <= 1) draws
+
+wisDistinctResamples :: IO Double
+wisDistinctResamples = do
+ draws <- take 2000 <$> wis 4 (sample uniform)
+ pure $ fromIntegral (length (nub draws))
+
+smokeWisWithinUnitInterval :: IO Bool
+smokeWisWithinUnitInterval = do
+ draws <- take 25 <$> wis 50 (sample uniform)
+ pure $ all (\x -> x >= 0 && x <= 1) draws
+
+-- Only the all-true sequence has weight (prior mass 1/1024). From weight 0 the chain accepts
+-- every move until it lands there, at sample index 408 with seed 1.
+zeroAcceptanceChecks :: IO [Bool]
+zeroAcceptanceChecks = do
+ let constrainedCoins = do
+ coins <- sample (replicateM 10 (bernoulli 0.5))
+ score (if and coins then 1 else 0)
+ pure coins
+ impossible = do x <- sample uniform; score 0; pure x
+ setStdGen (mkStdGen 1)
+ draws <- mh 0.5 constrainedCoins
+ escaped <- assertBool "mh reaches the support from a zero-weight initialization within 1024 transitions"
+ (any (and . fst) (take 1024 draws))
+ retained <- assertBool "mh rejects impossible proposals after entering the support"
+ (all (and . fst) (dropWhile (not . and . fst) (take 1024 draws)))
+ moving <- mapM (\(name, sampler) -> do
+ setStdGen (mkStdGen 1)
+ states <- take 16 <$> sampler impossible
+ assertBool (name ++ " accepts zero-to-zero initialization moves")
+ (length (nub (map fst states)) > 1 && all ((== 0) . getProduct . snd) states))
+ [("mh", mh 0.5), ("mhirreducible", mhirreducible 0.5 0.1)]
+ setStdGen (mkStdGen 1)
+ zeroBatch <- try (wis 16 impossible >>= evaluate . head) :: IO (Either ErrorCall Double)
+ noZeroNormalization <- assertBool "wis rejects an all-zero empirical measure"
+ (either (const True) (const False) zeroBatch)
+ setStdGen (mkStdGen 1)
+ positive <- take 32 <$> wis 256 (do
+ x <- sample uniform
+ score (if x > 0.5 then 1 else 0)
+ pure x)
+ positiveSupport <- assertBool "wis returns supported samples from a batch containing zero weights"
+ (all (>0.5) positive)
+ pure (escaped : retained : moving ++ [noZeroNormalization, positiveSupport])
+
+-- A fair coin b. On true, a uniform u is read and scored 6u, which integrates
+-- to 3; on false, the score is 1. So P(b) = 3/4 exactly, and the site u exists
+-- only on the true branch.
+coinModel :: Meas Bool
+coinModel = do
+ b <- sample (bernoulli 0.5)
+ if b then sample uniform >>= \u -> score (6 * u) else score 1
+ pure b
+
+-- The fraction of true draws is within five standard errors of 3/4. The
+-- standard error comes from the means of 40 consecutive batches, and it must be
+-- below 0.01 for the check to say anything.
+posteriorFraction :: String -> [Bool] -> IO Bool
+posteriorFraction name draws = do
+ let n = length draws
+ batches = 40
+ size = n `div` batches
+ batchMean k = fromIntegral (length (filter id (take size (drop (k * size) draws)))) / fromIntegral size :: Double
+ means = map batchMean [0 .. batches - 1]
+ p = sum means / fromIntegral batches
+ se = sqrt (sum [(m - p) ^ (2 :: Int) | m <- means] / fromIntegral (batches - 1) / fromIntegral batches)
+ ok = se < 0.01 && abs (p - 0.75) < 5 * se
+ putStrLn $ (if ok then "PASS " else "FAIL ") ++ name ++ ": P(true) = " ++ show p ++ ", standard error " ++ show se ++ ", exact 3/4"
+ pure ok
+
+-- wis n resamples from its n weighted particles, so each particle's share of
+-- the resamples is its weight over the total. With the same global seed,
+-- weightedsamples gives wis's particles, and the weights here are 1 and 3.
+wisResamplingCheck :: IO Bool
+wisResamplingCheck = do
+ let model = do x <- sample uniform; score (if x < 0.5 then 1 else 3); pure x
+ n = 5
+ draws = 20000 :: Int
+ setStdGen (mkStdGen 24)
+ particles <- take n <$> weightedsamples model
+ setStdGen (mkStdGen 24)
+ resamples <- take draws <$> wis n model
+ let total = sum [exp (ln w) | (_, w) <- particles]
+ shares = [ (fromIntegral (length (filter (== x) resamples)) / fromIntegral draws, exp (ln w) / total)
+ | (x, w) <- particles ]
+ within5 (observed, expected) = abs (observed - expected) <= 5 * sqrt (expected * (1 - expected) / fromIntegral draws)
+ assertBool ("wis 5 resamples each particle in proportion to its weight, (observed, expected) = " ++ show shares)
+ (all within5 shares && length (filter (`elem` map fst particles) resamples) == draws)
+
+posteriorChecks :: IO [Bool]
+posteriorChecks = do
+ setStdGen (mkStdGen 21)
+ chain <- map fst . take 40000 . drop 1000 <$> mh 0.5 coinModel
+ setStdGen (mkStdGen 22)
+ irreducible <- map fst . take 40000 . drop 1000 <$> mhirreducible 0.5 0.1 coinModel
+ sequence [posteriorFraction "mh 0.5 coin posterior" chain, posteriorFraction "mhirreducible 0.5 0.1 coin posterior" irreducible, wisResamplingCheck]
diff --git a/test/Api/Processes.hs b/test/Api/Processes.hs
new file mode 100644
--- /dev/null
+++ b/test/Api/Processes.hs
@@ -0,0 +1,214 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+-- The Wiener and memo checks need a fresh hidden table per call, which floating
+-- a constant random function out of its lambda would share.
+module Api.Processes where
+
+import Control.Exception (ErrorCall, SomeException, evaluate, try)
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import Numeric.SpecFunctions (erfc)
+import LazyPPL.Distributions
+import LazyPPL.Distributions.GP (gp, wiener)
+import LazyPPL.Distributions.IBP (ibp)
+import LazyPPL.Distributions.Memoization (memoize, memrec)
+import qualified LazyPPL.Distributions.DirichletP as DP
+import qualified LazyPPL.Distributions.Mondrian as Mondrian
+import System.Random (RandomGen(..), SplitGen(..), mkStdGen, setStdGen)
+import Test.QuickCheck (NonEmptyList(..), NonNegative(..), Positive(..), Property, ioProperty, isSuccess, quickCheckResult)
+
+import Api.Support
+
+memoizeMismatches :: Double
+memoizeMismatches =
+ let g = runProb (memoize (\n -> pure (n :: Int))) (randomTree (mkStdGen 1))
+ in fromIntegral (length (filter (\n -> g n /= n) [0 .. 40]))
+
+standardDeviation :: [Double] -> Double
+standardDeviation xs =
+ let n = fromIntegral (length xs)
+ m = sum xs / n
+ in sqrt (sum [(x - m) ^ (2 :: Int) | x <- xs] / (n - 1))
+
+-- Since cov 0 0 = 1, f 0 has standard deviation 1 up to the nugget.
+gpStandardDeviationAtZero :: Double
+gpStandardDeviationAtZero =
+ standardDeviation [runProb (gp (\x y -> exp (negate ((x - y) ^ (2 :: Int))))) (randomTree (mkStdGen s)) 0 | s <- [1 .. 400]]
+
+-- With a constant kernel c, f 1 - f 0 is only nugget noise, of variance 2*c*2^-33.
+gpConstantKernelGap :: Double -> IO Double
+gpConstantKernelGap c = do
+ let f = runProb (gp (\_ _ -> c)) (randomTree (mkStdGen 3))
+ r <- try (evaluate (f 0 `seq` abs (f 1 - f 0) / sqrt c)) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) id r)
+
+-- Scaling the kernel by s scales the draw by sqrt s, so even a huge s gives a finite draw.
+gpScaledKernelFinite :: Double -> IO Double
+gpScaledKernelFinite s = do
+ let f = runProb (gp (\x y -> s * exp (negate ((x - y) ^ (2 :: Int))))) (randomTree (mkStdGen 4))
+ r <- try (evaluate (sum [abs (f x) | x <- [0, 0.1 .. 3]] / sqrt s)) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) (\v -> if isNaN v || isInfinite v then 1 / 0 else 0) r)
+
+-- The rank-two kernel gives a line plus nugget noise. At x = 5 the residual has nugget
+-- variance 92*c*2^-33, the 92 in the bound.
+gpRankTwoLineResidual :: Double -> IO Double
+gpRankTwoLineResidual c = do
+ let f = runProb (gp (\x y -> c * (1 + x * y))) (randomTree (mkStdGen 4))
+ line x = f 0 + x * (f 1 - f 0)
+ r <- try (evaluate (maximum [abs (f x - line x) | x <- [0 .. 5]] / sqrt c)) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) id r)
+
+-- c cos (x - y) has rank two as well, and its draw must still be finite.
+gpCosineKernelFinite :: Double -> IO Double
+gpCosineKernelFinite c = do
+ let f = runProb (gp (\x y -> c * cos (x - y))) (randomTree (mkStdGen 5))
+ r <- try (evaluate (sum [abs (f x) | x <- [0 .. 5]] / sqrt c)) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) (\v -> if isNaN v || isInfinite v then 1 / 0 else 0) r)
+
+-- On the same random tree, gp (s * k) is sqrt s times gp k, nugget included.
+gpScaleEquivarianceError :: IO Double
+gpScaleEquivarianceError = do
+ let kernels = [\_ _ -> 1, \x y -> 1 + x * y, \x y -> cos (x - y), \x y -> exp (negate ((x - y) ^ (2 :: Int)))]
+ draw s k = runProb (gp (\x y -> s * k x y)) (randomTree (mkStdGen 6))
+ gap s k = let f1 = draw 1 k; fs = draw s k
+ in maximum [abs (fs x / sqrt s - f1 x) / (1 + abs (f1 x)) | x <- [0 .. 5]]
+ r <- try (evaluate (maximum [gap s k | k <- kernels,
+ s <- [1e-300,1e-100,1e8,1e16,1e100,1e300]])) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) id r)
+
+-- Even a variance of 2^-1074, the smallest positive double, gives the first point a draw, as
+-- there is no absolute cutoff. Later pivots may underflow.
+gpSmallestVarianceSample :: IO Double
+gpSmallestVarianceSample = do
+ let s = encodeFloat 1 (-1074)
+ f = runProb (gp (\_ _ -> s)) (randomTree (mkStdGen 4))
+ r <- try (evaluate (abs (f 0) / sqrt s)) :: IO (Either SomeException Double)
+ pure (either (const (1 / 0)) id r)
+
+-- The first customer takes Poisson(1) dishes, so the first row is empty with probability exp(-1).
+ibpFirstRowEmptyFraction :: Double
+ibpFirstRowEmptyFraction =
+ let firstRowEmpty s = case runProb (ibp 1) (randomTree (mkStdGen s)) of
+ row : _ -> null row
+ [] -> error "ibp returned no rows"
+ in fromIntegral (length (filter firstRowEmpty [1 .. 2000 :: Int])) / 2000
+
+smokeGpFinite :: IO Bool
+smokeGpFinite = do
+ let cov x y = exp (negate ((x - y) * (x - y)))
+ f = runProb (gp cov) (randomTree (mkStdGen 12345))
+ ys = [f 0, f 0.25, f 1.0]
+ pure $ all (\y -> not (isNaN y) && not (isInfinite y)) ys
+
+memoStochasticChecks :: IO [Bool]
+memoStochasticChecks = do
+ result <- quickCheckResult $ \seed (NonNegative key) -> ioProperty $ do
+ let f = runProb (memoize (const uniform :: Int -> Prob Double)) (randomTree (mkStdGen seed))
+ otherKey = if key == maxBound then 0 else key + 1
+ values <- mapM (evaluate . f) [key, otherKey, key]
+ pure $ case values of
+ [first, other, repeated] -> first == repeated && first /= other
+ _ -> False
+ sequence [assertBool "memoize reuses sampled values and distinguishes keys" (isSuccess result)]
+
+newtype OrdOnlyKey = OrdOnlyKey Int deriving (Eq, Ord)
+
+recursiveMemoCheck :: IO [Bool]
+recursiveMemoCheck = do
+ let f = runProb (memrec (\self (OrdOnlyKey n) ->
+ pure (if n == 0 then 1 else self (OrdOnlyKey (n-1)) + 1))) (randomTree (mkStdGen 1))
+ sequence [assertBool "memrec accepts keys with Ord and no Show" (f (OrdOnlyKey 3) == (4 :: Int))]
+
+dirichletProcessChecks :: IO [Bool]
+dirichletProcessChecks = do
+ result <- quickCheckResult $ \seed ->
+ let tree = randomTree (mkStdGen seed)
+ constantDraw = do distribution <- DP.dp 1 (pure (7 :: Int)); distribution
+ in runProb constantDraw tree == 7 && runProb (categorical (0:1:undefined)) tree == 1
+ sequence [assertBool "Dirichlet process preserves constant atoms and lazy categorical selection" (isSuccess result)]
+
+-- x = 1/2 itself goes to the right child. The blocks have bias 0 and 1, so a wrong traversal
+-- or likelihood shows.
+mondrianChecks :: IO [Bool]
+mondrianChecks = do
+ let box = [(0,1),(0,1)]
+ tree = Mondrian.Partition 0 0.5 box
+ (Mondrian.Block 0 [(0,0.5),(0,1)]) (Mondrian.Block 1 [(0.5,1),(0,1)])
+ draw seed p = runProb p (randomTree (mkStdGen seed))
+ expected = [(0.25,False),(0.5,True),(0.75,True)]
+ halfTree = let t = Tree 0.5 (repeat t) in t
+ -- With every label 1/2, the root (cost log(2)/2 < 0.4) is cut along dimension 1 at 1/2,
+ -- and each child (cost log(2)/1.5) exceeds the budget left.
+ expectedPrior = Mondrian.Partition 1 0.5 box
+ (Mondrian.Block 0.25 [(0,1),(0,0.5)]) (Mondrian.Block 0.25 [(0,1),(0.5,1)])
+ result <- quickCheckResult $ \seed -> and
+ [ Mondrian.likelihoodFromMondrian2D tree x y value == 1 &&
+ Mondrian.likelihoodFromMondrian2D tree x y (not value) == 0 &&
+ draw seed (Mondrian.sampleFromMondrian2D tree x y) == value
+ | (x,value) <- expected, y <- [0,1] ]
+ prior <- quickCheckResult $ \seed ->
+ draw seed (Mondrian.randomMondrian (pure (7 :: Int)) 0 box) == Mondrian.Block 7 box
+ sequence
+ [ assertBool "Mondrian likelihood and samples follow the cut" (isSuccess result)
+ , assertBool "zero-budget Mondrian draws its base without cuts" (isSuccess prior)
+ , assertBool "Mondrian samples the independently specified first cut and child boxes"
+ (runProb (Mondrian.randomMondrian (pure 0.25) 0.4 box) halfTree == expectedPrior)
+ ]
+
+-- The Wiener bridge at unequal gaps. Given W(0) = 0 and W(T) = w, W(s) for
+-- 0 < s < T is normal with mean (s/T) w and standard deviation
+-- sqrt(s (T - s) / T), representable here even where s/T or s (T - s) / T is
+-- not. On the tree whose every label is 1/4, every standard normal draw is the
+-- lower quartile z = -0.6744897501960817, so w = sqrt T z and W(s) has an exact
+-- value. Over seeds, the standardized W(s) is standard normal; that law is
+-- checked with the Kolmogorov-Smirnov threshold of lawChecks.
+wienerBridgeChecks :: IO [Bool]
+wienerBridgeChecks = do
+ let z = -0.6744897501960817
+ leastPositive = encodeFloat 1 (-1074) :: Double
+ afterT s t tree = let f = runProb wiener tree in f t `seq` (f t, f s)
+ exactAt s t = do
+ let expected = z * sqrt s * sqrt ((t - s) / t) + s * (z / sqrt t)
+ r <- within 500000 (snd (afterT s t (constantLabel 0.25)))
+ assertBool ("wiener at " ++ show s ++ " after " ++ show t ++ " on labels 1/4 is " ++ show expected
+ ++ ", got " ++ maybe "an error or no answer" show r)
+ (maybe False (\v -> abs (v - expected) <= 1e-12 * abs expected) r)
+ n = 10000 :: Int
+ threshold = sqrt (log (2 / 1e-6) / (2 * fromIntegral n))
+ standardNormalCdf x = erfc (- x / sqrt 2) / 2
+ lawAt s t = do
+ let residual seed = let (w, v) = afterT s t (randomTree (mkStdGen seed))
+ in (v - s * (w / t)) / (sqrt s * sqrt ((t - s) / t))
+ d = ksDistance standardNormalCdf (map residual [1 .. n])
+ assertBool ("wiener at " ++ show s ++ " after " ++ show t ++ ": standardized draws at distance " ++ show d
+ ++ " from the standard normal (threshold " ++ show threshold ++ ")")
+ (d <= threshold)
+ sequence
+ [ exactAt 1e-200 1e200
+ , exactAt leastPositive (2 * leastPositive)
+ , exactAt 1 4
+ , lawAt 1e-200 1e200
+ , lawAt leastPositive (2 * leastPositive)
+ , lawAt 1 4
+ ]
+
+-- Given f(2s), the Wiener bridge at s has mean f(2s)/2 and variance s/2, so on
+-- one tree f(s)/sqrt(s) does not depend on s. A query at an invalid time is an
+-- error and leaves the process unchanged.
+processChecks :: IO [Bool]
+processChecks = do
+ let scaled s = let f = runProb wiener (randomTree (mkStdGen 13)) in f (2 * s) `seq` f s / sqrt s
+ reference = scaled 1
+ f = runProb wiener (randomTree (mkStdGen 14))
+ scales <- sequence
+ [ do value <- within 500000 (scaled s)
+ assertBool ("wiener f(s)/sqrt(s) at s = " ++ show s ++ " is " ++ show reference ++ ", got " ++ show value)
+ (maybe False (\v -> abs (v - reference) <= 1e-12 * abs reference) value)
+ | s <- [1e-200, 1e-100, 1e100, 1e200] ]
+ invalid <- try (evaluate (f (0/0)))
+ after <- within 500000 (f 1)
+ atomic <- sequence
+ [ assertBool ("wiener rejects time NaN" ++ either (const "") (\v -> ", got " ++ show (v :: Double)) invalid)
+ (either (\e -> const True (e :: ErrorCall)) (const False) invalid)
+ , assertBool ("wiener after a rejected query gives a finite f 1, got " ++ show after)
+ (maybe False (\v -> not (isNaN v || isInfinite v)) after)
+ ]
+ pure (scales ++ atomic)
diff --git a/test/Api/Support.hs b/test/Api/Support.hs
new file mode 100644
--- /dev/null
+++ b/test/Api/Support.hs
@@ -0,0 +1,105 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Api.Support where
+
+import Control.Exception (ErrorCall, SomeException, evaluate, try)
+import Data.List (isInfixOf, isPrefixOf, nub, sort, tails)
+import LazyPPL (Meas(..), Prob, Tree(..), mh, mhirreducible, randomTree, runProb, sample, score, scoreLog, scoreProductLog, uniform, weightedsamples, wis)
+import Numeric.MathFunctions.Comparison (addUlps)
+import System.Timeout (timeout)
+import LazyPPL.Distributions
+
+approxEq :: String -> Double -> Double -> Double -> IO Bool
+approxEq name expected actual tol = do
+ let ok = abs (expected - actual) < tol
+ putStrLn $
+ (if ok then "PASS" else "FAIL")
+ ++ " "
+ ++ name
+ ++ " expected="
+ ++ show expected
+ ++ " actual="
+ ++ show actual
+ pure ok
+
+assertBool :: String -> Bool -> IO Bool
+assertBool name ok = do
+ putStrLn $ (if ok then "PASS" else "FAIL") ++ " " ++ name
+ pure ok
+
+-- Evaluates a value with a time limit. An exception or a timeout gives Nothing.
+within :: Int -> a -> IO (Maybe a)
+within micros x = do
+ result <- try (timeout micros (evaluate x))
+ pure $ case result of
+ Left err -> const Nothing (err :: SomeException)
+ Right value -> value
+
+-- Passes when evaluating the value raises an error call.
+rejects :: String -> a -> IO Bool
+rejects name x = do
+ result <- try (timeout 500000 (evaluate x))
+ assertBool (name ++ " is rejected" ++ case result of
+ Left err -> const "" (err :: ErrorCall)
+ Right Nothing -> " (no answer within 0.5 s)"
+ Right (Just _) -> " (it returned a value)")
+ (either (const True) (const False) result)
+
+-- Passes when running the action raises an error call.
+rejectsIO :: String -> IO Double -> IO Bool
+rejectsIO name action = do
+ result <- try action
+ assertBool (name ++ case result of
+ Left err -> const "" (err :: ErrorCall)
+ Right value -> " (it returned " ++ show value ++ ")")
+ (either (const True) (const False) result)
+
+-- The tree whose every label is u.
+constantLabel :: Double -> Tree
+constantLabel u = let t = Tree u (repeat t) in t
+
+-- The largest distance between the empirical distribution function of the
+-- sample and cdf, with an allowance for rounding: at a draw x the exact function
+-- is taken at x (at the least positive double for a draw that underflowed to
+-- 0), and just below x at the double before x. A draw is a rounded double, so
+-- equal draws can be an atom: Beta(0.001,1) puts 0.475 below the least positive
+-- double, which all round to 0. statistics-0.16.5.0's kolmogorovSmirnovCdfD
+-- computes the statistic for a continuous law, without such atoms.
+ksDistance :: (Double -> Double) -> [Double] -> Double
+ksDistance cdf xs = maximum
+ [ max (i / n - cdf (if x == 0 then encodeFloat 1 (-1074) else x)) (cdf (addUlps (-1) x) - (i - 1) / n)
+ | (i, x) <- zip [1 ..] (sort xs) ]
+ where n = fromIntegral (length xs)
+
+-- The rows of a table that test/reference/reference.py wrote with mpmath, each
+-- split at its tabs; lines that start with # are comments. Cabal runs the test
+-- in the package root, where the path below starts.
+referenceTable :: FilePath -> IO [[String]]
+referenceTable name = do
+ text <- readFile ("test/reference/" ++ name)
+ pure [cells line | line <- lines text, not (null line), take 1 line /= "#"]
+ where
+ cells s = case break (== '\t') s of
+ (cell, []) -> [cell]
+ (cell, _ : rest) -> cell : cells rest
+
+-- Prints the failures of a table check and one summary line.
+tableReport :: String -> [(Bool, String)] -> IO [Bool]
+tableReport name results = do
+ mapM_ (\(_, detail) -> putStrLn ("FAIL " ++ detail)) (filter (not . fst) results)
+ let passed = length (filter fst results)
+ putStrLn ((if passed == length results then "PASS " else "FAIL ") ++ name ++ ": "
+ ++ show passed ++ " of " ++ show (length results) ++ " rows")
+ pure (map fst results)
+
+-- Whether evaluating the value raises an error call within 2 s.
+raisesError :: a -> IO Bool
+raisesError x = do
+ result <- try (timeout 2000000 (evaluate x))
+ pure (either (\err -> const True (err :: ErrorCall)) (const False) result)
+
+raisesBetaLabel :: a -> IO Bool
+raisesBetaLabel x = do
+ result <- try (timeout 2000000 (evaluate x))
+ pure (check result)
+ where check :: Either BetaLabelError b -> Bool
+ check = either (const True) (const False)
diff --git a/test/Chain.hs b/test/Chain.hs
new file mode 100644
--- /dev/null
+++ b/test/Chain.hs
@@ -0,0 +1,108 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Main where
+
+import Control.DeepSeq (force)
+import Control.Exception (SomeException, evaluate, try)
+import Control.Monad (liftM, unless, forM, forM_)
+import Control.Monad.State.Lazy (State, get, put, runState)
+import Control.Monad.Trans.Writer (runWriterT)
+import Data.Monoid (Product(..))
+import GHC.Float (castDoubleToWord64)
+import LazyPPL
+import LazyPPL.Internal.Sampler (randomLabel)
+import Numeric.Log (Log, ln)
+import System.Random hiding (uniform)
+import Test.QuickCheck hiding (sample)
+import Test.QuickCheck.Random (mkQCGen)
+
+-- The tree-keeping transition and lazy State traversal are from
+-- 8168aff:src/LazyPPL.hs. They do not use the replay history or checkpoints.
+reference :: forall a. Double -> Maybe Double -> Meas a -> IO [(a, Product (Log Double))]
+reference p refresh (Meas m) = do
+ g <- newStdGen
+ let (g1,g2) = split g
+ t = randomTree g1
+ (x,w) = runProb (runWriterT m) t
+ (samples,_) = runState (iterateM step (t,x,w)) g2
+ pure (map (\(_,v,weight) -> (v,weight)) samples)
+ where
+ step :: (Tree,a,Product (Log Double)) -> State StdGen (Tree,a,Product (Log Double))
+ step (t,x,w) = do
+ g <- get
+ let (g1,g2) = split g
+ t' = case refresh of
+ Nothing -> mutate p g1 t
+ Just q -> let (r,g1') = random g1
+ in if r < q then randomTree g1' else mutate p g1' t
+ (x',w') = runProb (runWriterT m) t'
+ ratio = getProduct w' / getProduct w
+ (r,g2') = random g2
+ put g2'
+ if r < min 1 (exp (ln ratio)) then pure (t',x',w') else pure (t,x,w)
+ iterateM f x = do
+ x' <- f x
+ (x':) `liftM` iterateM f x'
+
+mutate :: Double -> StdGen -> Tree -> Tree
+mutate p g (Tree a ts) =
+ let (a',g') = randomLabel g
+ (a'',g'') = randomLabel g'
+ children seed (t:rest) = let (g1,g2) = split seed
+ in mutate p g1 t : children g2 rest
+ children _ [] = []
+ in Tree (if a' < p then a'' else a) (children g'' ts)
+
+at :: [Int] -> Tree -> Double
+at [] (Tree x _) = x
+at (i:is) (Tree _ ts) = at is (ts !! i)
+
+model :: Int -> Meas ([Int] -> Double)
+model kind = do
+ tree <- sample (Prob id)
+ let root = at [] tree
+ branch = if root < 0.5 then [0,2] else [2,0]
+ case kind of
+ 0 -> pure ()
+ 1 -> score (if root < 0.3 then 1 else 1e-12)
+ _ -> score (0.01 + at branch tree)
+ pure (\path -> if path == [99] then error "delayed payload" else at path tree)
+
+chainProperty :: Property
+chainProperty = forAll arbitrary $ \seed ->
+ forAll (elements [0,0.001,0.2,1]) $ \rate ->
+ forAll (elements [Nothing,Just 0.05,Just 1]) $ \refresh ->
+ forAll (chooseInt (0,2)) $ \kind ->
+ forAll (elements [33,97,257,4101]) $ \count -> ioProperty $ do
+ let measure = model kind
+ sampler = maybe (mh rate) (mhirreducible rate) refresh
+ paths = [[],[0],[1,0],[0,2],[2,0],[5,3,1]]
+ observe (f,w) = map (castDoubleToWord64 . f) paths ++ [castDoubleToWord64 (ln (getProduct w))]
+ setStdGen (mkStdGen seed)
+ actual <- sampler measure
+ setStdGen (mkStdGen seed)
+ expected <- reference rate refresh measure
+ -- Advance before demanding the old payloads, including across seed-store growth.
+ _ <- evaluate (snd (actual !! (count-1)))
+ _ <- evaluate (snd (expected !! (count-1)))
+ let indices = [0 .. min 256 (count-1)] ++ [count-1]
+ observed <- evaluate (force (map (observe . (actual !!)) indices))
+ oracle <- evaluate (force (map (observe . (expected !!)) indices))
+ failures <- forM [actual,expected] $ \xs -> do
+ outcome <- try (evaluate (fst (head xs) [99])) :: IO (Either SomeException Double)
+ pure (either (const True) (const False) outcome)
+ pure $ counterexample (show (seed,rate,refresh,kind,count)) $
+ observed === oracle .&&. failures === [True,True]
+
+main :: IO ()
+main = do
+ forM_ [Nothing, Just 0.05, Just 1] $ \refresh -> do
+ let sampler = maybe (mh 0.2) (mhirreducible 0.2) refresh
+ forM_ [sampler, reference 0.2 refresh] $ \run -> do
+ xs <- run (pure (error "undemanded payload" :: Bool))
+ count <- evaluate (length (take 128 xs))
+ unless (count == 128) (fail "Sampler forced an undemanded payload")
+ bad <- run (score (error "demanded score") >> pure True)
+ outcome <- try (evaluate (head bad)) :: IO (Either SomeException (Bool,Product (Log Double)))
+ unless (either (const True) (const False) outcome) (fail "Sampler hid a score exception")
+ result <- quickCheckWithResult stdArgs {maxSuccess=200, replay=Just (mkQCGen 20261026,0)} chainProperty
+ unless (isSuccess result) (fail "Seeded chains differ from the tree-keeping sampler")
diff --git a/test/Concurrency.hs b/test/Concurrency.hs
new file mode 100644
--- /dev/null
+++ b/test/Concurrency.hs
@@ -0,0 +1,143 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+module Main where
+
+import Control.Concurrent (forkFinally, newEmptyMVar, putMVar, takeMVar)
+import Control.Concurrent.MVar (MVar)
+import Control.Exception (evaluate, throwIO)
+import Control.Monad (forM, replicateM, unless, when)
+import Data.IORef (IORef, atomicModifyIORef', newIORef, writeIORef)
+import qualified Data.Set as Set
+import LazyPPL (Prob, mh, randomTree, runProb, sample, uniform, weightedsamples)
+import LazyPPL.Distributions.Counter (newCounter, readAndIncrement)
+import LazyPPL.Distributions.GP (gp)
+import LazyPPL.Distributions.Memoization (generalmemoize)
+import System.Exit (exitFailure)
+import System.IO.Unsafe (unsafePerformIO)
+import System.Random (mkStdGen)
+import System.Timeout (timeout)
+
+-- Runs `draw` `runs` times in each of two threads at once.
+concurrentDraws :: Int -> IO a -> IO [a]
+concurrentDraws runs draw = do
+ slots <- forM [1, 2 :: Int] $ \_ -> do
+ slot <- newEmptyMVar
+ _ <- forkFinally (replicateM runs (draw >>= evaluate)) (putMVar slot)
+ pure slot
+ concat <$> mapM (\slot -> takeMVar slot >>= either throwIO pure) slots
+
+report :: String -> Bool -> String -> IO Bool
+report name ok detail = do
+ putStrLn $ (if ok then "PASS " else "FAIL ") ++ name ++ ": " ++ detail
+ pure ok
+
+-- Each inference run takes its own generator from the global one. Two runs
+-- that start at the same moment must still get different generators, so no
+-- two runs share their first draw.
+distinctFirstDraws :: String -> IO Double -> IO Bool
+distinctFirstDraws name draw = do
+ draws <- concurrentDraws 20000 draw
+ let distinct = Set.size (Set.fromList draws)
+ report name (distinct == length draws)
+ (show distinct ++ " distinct first draws in " ++ show (length draws) ++ " concurrent runs")
+
+-- The pause point of the interleaving tests below: the thread that makes the
+-- `pauseAt`-th marked call signals `inside`, then waits for `release`. The
+-- wait has a time limit, because a correct implementation can make the other
+-- thread wait for this one.
+{-# NOINLINE pauseHook #-}
+pauseHook :: IORef (IO ())
+pauseHook = unsafePerformIO (newIORef (pure ()))
+
+markedCall :: IO ()
+markedCall = do
+ action <- atomicModifyIORef' pauseHook (\a -> (a, a))
+ action
+
+pauseAt :: Int -> IO (MVar (), MVar ())
+pauseAt target = do
+ inside <- newEmptyMVar
+ release <- newEmptyMVar
+ calls <- newIORef (0 :: Int)
+ writeIORef pauseHook $ do
+ n <- atomicModifyIORef' calls (\c -> (c + 1, c + 1))
+ when (n == target) $ do
+ putMVar inside ()
+ _ <- timeout 2000000 (takeMVar release)
+ pure ()
+ pure (inside, release)
+
+-- Keys whose comparison makes a marked call when the left key is 1. The
+-- comparison results are the ordinary ones.
+newtype Key = Key Int deriving (Eq, Show)
+
+instance Ord Key where
+ compare (Key a) (Key b) = unsafePerformIO (when (a == 1) markedCall) `seq` compare a b
+
+-- Two threads query a memoized stochastic function at two new keys at once.
+-- The first thread pauses at its second comparison, after it has read the
+-- table and before it has stored its entry, and the second thread runs in
+-- between. Each key must still get its own draw, and a repeated query must
+-- return the draw made for its key.
+memoInterleaving :: IO Bool
+memoInterleaving = do
+ let f = runProb (generalmemoize (const uniform :: Key -> Prob Double)) (randomTree (mkStdGen 7))
+ _ <- evaluate (f (Key 0))
+ (inside, release) <- pauseAt 2
+ slot <- newEmptyMVar
+ _ <- forkFinally (evaluate (f (Key 1))) (putMVar slot)
+ takeMVar inside
+ second <- evaluate (f (Key 2))
+ putMVar release ()
+ first <- takeMVar slot >>= either throwIO pure
+ writeIORef pauseHook (pure ())
+ repeated1 <- evaluate (f (Key 1))
+ repeated2 <- evaluate (f (Key 2))
+ report "memoize under an interleaving" (first /= second && repeated1 == first && repeated2 == second)
+ ("(first1,first2,repeat1,repeat2) = " ++ show (first, second, repeated1, repeated2))
+
+-- The same interleaving for a white-noise Gaussian process, whose values at
+-- distinct points are independent. The covariance function makes the marked
+-- call at (1,1), inside the first thread's query.
+gpInterleaving :: IO Bool
+gpInterleaving = do
+ let cov x y = unsafePerformIO (when (x == 1 && y == 1) markedCall) `seq` (if x == y then 1 else 0)
+ f = runProb (gp cov) (randomTree (mkStdGen 8))
+ _ <- evaluate (f 0)
+ (inside, release) <- pauseAt 1
+ slot <- newEmptyMVar
+ _ <- forkFinally (evaluate (f 1)) (putMVar slot)
+ takeMVar inside
+ second <- evaluate (f 2)
+ putMVar release ()
+ first <- takeMVar slot >>= either throwIO pure
+ writeIORef pauseHook (pure ())
+ repeated1 <- evaluate (f 1)
+ repeated2 <- evaluate (f 2)
+ report "white-noise gp under an interleaving" (first /= second && repeated1 == first && repeated2 == second)
+ ("(f1,f2,repeat f1,repeat f2) = " ++ show (first, second, repeated1, repeated2))
+
+-- A counter hands out each number once, even to two threads at once. The
+-- counter is bound by evaluate, as a Prob bind would bind it: a let-bound
+-- counter can be inlined into the lambda below, with one IORef per call.
+counterUnique :: IO Bool
+counterUnique = do
+ counter <- evaluate (runProb newCounter (randomTree (mkStdGen 9)))
+ let next k = runProb (readAndIncrement counter) (randomTree (mkStdGen k))
+ seeds <- newIORef (0 :: Int)
+ numbers <- concurrentDraws 20000 $ do
+ k <- atomicModifyIORef' seeds (\s -> (s + 1, s))
+ evaluate (next k)
+ let distinct = Set.size (Set.fromList numbers)
+ report "counter shared by two threads" (distinct == length numbers)
+ (show distinct ++ " distinct numbers from " ++ show (length numbers) ++ " increments")
+
+main :: IO ()
+main = do
+ results <- sequence
+ [ distinctFirstDraws "weightedsamples" (fst . head <$> weightedsamples (sample uniform))
+ , distinctFirstDraws "mh" (fst . head <$> mh 1 (sample uniform))
+ , memoInterleaving
+ , gpInterleaving
+ , counterUnique
+ ]
+ unless (and results) exitFailure
diff --git a/test/GaussianCondition.hs b/test/GaussianCondition.hs
new file mode 100644
--- /dev/null
+++ b/test/GaussianCondition.hs
@@ -0,0 +1,237 @@
+module Main where
+
+import Control.Exception (SomeException, displayException, evaluate, try)
+import Control.Monad (forM, forM_, unless)
+import Data.List (isPrefixOf, transpose)
+import LazyPPL (Tree(..), randomTree, runProb)
+import LazyPPL.Distributions (normal)
+import LazyPPL.Distributions.GP (gp, wiener)
+import LazyPPL.Distributions.GP.Pure (emptyGPState, sampleGP)
+import System.Environment (getArgs)
+import System.Random (mkStdGen)
+import Test.QuickCheck
+import Test.QuickCheck.Random (mkQCGen)
+
+finite :: Double -> Bool
+finite x = not (isNaN x || isInfinite x)
+
+close :: Double -> Double -> Double -> Bool
+close tolerance actual expected =
+ finite actual && abs (actual - expected) <= tolerance
+
+-- 2^-33 * cov x x is added to every variance, the first one included.
+nugget :: Double
+nugget = 2 ** (-33)
+
+-- Runs gp's own sampler, sampleGP, on the draws zs we choose.
+samplePath :: (Double -> Double -> Double) -> [Double] -> [Double] -> [Double]
+samplePath cov = go emptyGPState
+ where
+ go _ [] [] = []
+ go state (x:xs) (z:zs) =
+ let (value,next) = sampleGP cov state x z
+ in value : go next xs zs
+ go _ _ _ = error "test point/innovation lengths differ"
+
+-- Unit vectors give the columns of L (f = L z). L L^T should equal the kernel plus nugget, so no
+-- second Cholesky is needed.
+covarianceResidual :: [[Double]] -> Double
+covarianceResidual kernel =
+ let n = length kernel
+ xs = map fromIntegral [0..n-1]
+ cov x y = kernel !! round x !! round y
+ columns = [samplePath cov xs [if j == k then 1 else 0 | j <- [0..n-1]]
+ | k <- [0..n-1]]
+ rows = transpose columns
+ target = [[v + (if i == j then nugget*v else 0)
+ | (j,v) <- zip [0::Int ..] row] | (i,row) <- zip [0..] kernel]
+ actual = [[sum (zipWith (*) a b) | b <- rows] | a <- rows]
+ norm values = sqrt (sum [v*v | row <- values, v <- row])
+ in norm (zipWith (zipWith (-)) target actual) / norm target
+
+-- Closed-form 2-by-2 Cholesky factor, so the reference needs no solver.
+twoByTwo :: Property
+twoByTwo = forAll (vectorOf 5 (choose (-0.5,0.5))) $ \values ->
+ case values of
+ [x,b,y,z,w] ->
+ let a = 1+abs x; c = 1+abs y; q = 1+nugget
+ cov i j = if i == j then (if i == 0 then a else c) else b
+ actual = samplePath cov [0,1] [z,w]
+ expected = [sqrt (a*q)*z, b/sqrt (a*q)*z + sqrt (c*q-b*b/(a*q))*w]
+ in counterexample (show (actual,expected)) $
+ and (zipWith (close 1e-12) actual expected)
+ _ -> property False
+
+-- A Householder reflection (https://en.wikipedia.org/wiki/Householder_transformation) gives
+-- kernels with known eigenvalues, down to 1e-17 and 0. L L^T must reproduce
+-- kernel + nugget*diag(kernel), dropping no mode.
+knownSpectrum :: Property
+knownSpectrum = forAll (chooseInt (3,10)) $ \n ->
+ forAll (vectorOf n (choose (0.2,1))) $ \axis ->
+ forAll (elements [0,1,2::Int]) $ \kind ->
+ let norm = sum (map (\x -> x*x) axis)
+ q = [[(if i == j then 1 else 0)-2*a*b/norm
+ | (j,b) <- zip [0..] axis] | (i,a) <- zip [0::Int ..] axis]
+ eigenvalues = 2 : replicate (n-1) (case kind of 0 -> 0.5; 1 -> 1e-17; _ -> 0)
+ kernel = [[sum (zipWith3 (\d a b -> d*a*b) eigenvalues row col)
+ | col <- q] | row <- q]
+ residual = covarianceResidual kernel
+ -- 64*n*eps covers dot-product rounding and Cholesky's usual n-dependent error constant.
+ bound = 64 * fromIntegral n * encodeFloat 1 (-52)
+ in counterexample (show (kind,residual,bound)) (finite residual && residual <= bound)
+
+residual :: Property
+residual = forAll (chooseInt (2,10)) $ \n ->
+ forAll (vectorOf n (vectorOf n (choose (-1,1)))) $ \a ->
+ let kernel = [[sum (zipWith (*) row col)+(if i == j then 1 else 0)
+ | (j,col) <- zip [0..] a] | (i,row) <- zip [0::Int ..] a]
+ r = covarianceResidual kernel
+ in counterexample (show r) (finite r && r <= 64*fromIntegral n*encodeFloat 1 (-52))
+
+-- A diagonal covariance s gives independent draws with standard deviation sqrt(s*(1+nugget)).
+scaleCase :: [Double] -> Property
+scaleCase scales = forAll (elements scales) $ \s ->
+ forAll (vectorOf 4 (choose (-2,2))) $ \zs ->
+ let cov x y = if x == y then s else 0
+ actual = map (/sqrt s) (samplePath cov [0..3] zs)
+ expected = map (*(sqrt (1+nugget))) zs
+ in counterexample (show (s,actual,expected)) $
+ and (zipWith (close 1e-12) actual expected)
+
+smallScale, largeScale :: Property
+smallScale = scaleCase [1e-160,1e-200,1e-250]
+largeScale = scaleCase [1e8,1e160,1e200,1e250]
+
+-- gp uses its normal draws in order of first query, so we force each value before asking for
+-- the next.
+draw :: (Double -> Double -> Double) -> [Double] -> Int -> IO [Double]
+draw cov order seed = do
+ let f = runProb (gp cov) (randomTree (mkStdGen seed))
+ mapM (evaluate . f) order
+
+-- Both query orders must match the specified law and each other. For zero-mean Gaussians the
+-- standard error of a mean of X_i X_j is sqrt((Cii*Cjj+Cij^2)/N), and six of them keep false
+-- failures rare.
+orderIndependence :: Property
+orderIndependence = forAll (choose (-0.8,0.8)) $ \rho ->
+ forAll (chooseInt (1,1000000)) $ \firstSeed -> ioProperty $ do
+ let cov x y = if x == y then (if x == 0 then 1 else 2) else rho
+ count = 8192
+ q = 1+nugget
+ expected = [q,rho,2*q]
+ errors = [sqrt (2*q*q/fromIntegral count),
+ sqrt ((2*q*q+rho*rho)/fromIntegral count),
+ sqrt (8*q*q/fromIntegral count)]
+ moments pairs = [sum [a*a | [a,_] <- pairs]/fromIntegral count,
+ sum [a*b | [a,b] <- pairs]/fromIntegral count,
+ sum [b*b | [_,b] <- pairs]/fromIntegral count]
+ forward <- mapM (draw cov [0,1]) [firstSeed..firstSeed+count-1]
+ backward <- mapM (fmap reverse . draw cov [1,0]) [firstSeed+count..firstSeed+2*count-1]
+ let a = moments forward; b = moments backward
+ matches xs = and (zipWith3 (\v target se -> close (6*se) v target) xs expected errors)
+ agrees = and (zipWith3 (\v w se -> close (6*sqrt 2*se) v w) a b errors)
+ pure $ counterexample (show (rho,a,b,expected,errors)) (matches a && matches b && agrees)
+
+expectError :: String -> (Double -> Double -> Double) -> [Double] -> IO ()
+expectError name cov order = do
+ outcome <- try (draw cov order 1) :: IO (Either SomeException [Double])
+ case outcome of
+ Left err | name `isPrefixOf` displayException err -> pure ()
+ _ -> fail ("Expected " ++ name ++ ", got " ++ show outcome)
+
+invalidCovariance :: IO ()
+invalidCovariance = expectError "GPCovarianceNotPositiveDefinite"
+ (\x y -> if x == y then 100 else 100.3) [0,1]
+
+namedErrors :: IO ()
+namedErrors = do
+ invalidCovariance
+ expectError "GPCovarianceNotPositiveDefinite" (\_ _ -> -1) [0]
+ forM_ [0/0,1/0,negate (1/0)] $ \bad -> do
+ expectError "GPCovarianceNotFinite" (\_ _ -> bad) [0]
+ -- Only cov 0 1 is bad, an entry the factor never reads, yet the error must fire.
+ expectError "GPCovarianceNotFinite" (\x y -> if x == 0 && y == 1 then bad else 1) [0,1]
+ let zeroCross x y = if x == y then (if x == 0 then 0 else 1) else 0.5
+ twoZeros x y = if x == y then 0 else 0.5
+ forM_ [[0,1],[1,0]] $ \order -> do
+ expectError "GPCovarianceNotPositiveDefinite" zeroCross order
+ expectError "GPCovarianceNotPositiveDefinite" twoZeros order
+
+wienerBridge :: IO ()
+wienerBridge = forM_ (zip [20261029,20261030] [[2,1,3],[-2,-1,-3]]) $ \(seed,order) -> do
+ -- Each path gets its own tree, so the compiler cannot share one runProb wiener expression
+ -- and its memo table.
+ let tree = randomTree (mkStdGen seed)
+ f = runProb wiener tree
+ -- The table starts at (0,0). A new point takes its normal draw from child 1 + table size
+ -- of the input tree.
+ z i = case tree of
+ Tree _ children -> runProb (normal 0 1) (children !! i)
+ go _ [] = pure ()
+ go table (x:xs) = do
+ let lower = filter ((x) . fst) table
+ (mu,sd) = case (lower,upper) of
+ ([],_) -> let (b,v) = minimum upper in (v,sqrt (b-x))
+ (_,[]) -> let (a,v) = maximum lower in (v,sqrt (x-a))
+ _ -> let (a,v) = maximum lower; (b,w) = minimum upper
+ in (v+(x-a)*(w-v)/(b-a),sqrt ((b-x)*(x-a)/(b-a)))
+ expected = mu+sd*z (1+length table)
+ actual <- evaluate (f x)
+ unless (close 1e-12 actual expected) (fail (show (x,actual,expected)))
+ go ((x,actual):table) xs
+ go [(0,0)] order
+
+runProperty :: Int -> Property -> IO ()
+runProperty count prop = do
+ result <- quickCheckWithResult stdArgs {maxSuccess=count,replay=Just(mkQCGen 20261029,0)} prop
+ unless (isSuccess result) (fail "Gaussian process property failed")
+
+allChecks :: IO ()
+allChecks = do
+ namedErrors
+ zeros <- draw (\_ _ -> 0) [0,1,-1,2] 1
+ unless (zeros == replicate 4 0) (fail ("Zero covariance: " ++ show zeros))
+ -- Covariance 1 - 2^-53 leaves the kernel nearly singular, yet the second pivot squared stays
+ -- at q - a^2/q, about 2^-32.
+ let a = 1-encodeFloat 1 (-53)
+ tinyMode x y = if x == y then 1 else a
+ [_,innovation] = samplePath tinyMode [0,1] [0,1]
+ q = toRational (1+nugget)
+ expectedV = fromRational (q-toRational a*toRational a/q)
+ unless (close (1e-5*expectedV) (innovation*innovation) expectedV) $
+ fail ("Small spectral component: " ++ show (innovation*innovation,expectedV))
+ -- Close points of the constant kernel have conditional variance q - 1/q, but the same key
+ -- gives its stored value.
+ let [_,near] = samplePath (\_ _ -> 1) [0,1e-6] [0,1]
+ nearVariance = fromRational (q-1/q)
+ unless (close (1e-5*nearVariance) (near*near) nearVariance) $
+ fail ("Nearby re-observation: " ++ show (near*near,nearVariance))
+ let f = runProb (gp (\_ _ -> 1)) (randomTree (mkStdGen 9))
+ first <- evaluate (f 0)
+ _ <- evaluate (f 1e-6)
+ again <- evaluate (f 0)
+ unless (first == again) (fail "Repeated key changed")
+ grid <- draw (\x y -> exp (negate ((x-y)^ (2::Int))/0.1))
+ [fromIntegral i*0.02 | i <- [0..150::Int]] 7
+ unless (length grid == 151 && all finite grid) (fail "RBF grid failed")
+ wienerBridge
+ runProperty 512 twoByTwo
+ runProperty 100 knownSpectrum
+ runProperty 100 residual
+ runProperty 100 smallScale
+ runProperty 100 largeScale
+ runProperty 24 orderIndependence
+
+main :: IO ()
+main = do
+ args <- getArgs
+ case args of
+ [] -> allChecks
+ ["invalid"] -> invalidCovariance
+ ["spectrum"] -> runProperty 512 knownSpectrum
+ ["small"] -> runProperty 100 smallScale
+ ["large"] -> runProperty 100 largeScale
+ ["order"] -> runProperty 24 orderIndependence
+ ["bridge"] -> wienerBridge
+ _ -> fail "Expected no arguments, invalid, spectrum, small, large or order"
diff --git a/test/HiddenState.hs b/test/HiddenState.hs
new file mode 100644
--- /dev/null
+++ b/test/HiddenState.hs
@@ -0,0 +1,101 @@
+module Main where
+
+-- The documented use of the values with hidden state that runProb returns:
+-- counters, IBP restaurants, generalmemoize functions, and the random functions
+-- of wiener and gp. Each check binds the value with evaluate before an IO action
+-- that runs once per query, and requires the answers that the same queries give
+-- inside one Prob computation on the same tree. This module takes the compiler's
+-- default options; hidden-state-o2-test compiles it at -O2.
+
+import Control.Exception (evaluate)
+import Control.Monad (replicateM, unless)
+import Data.IORef (atomicModifyIORef', newIORef)
+import Data.List (nub)
+import LazyPPL (Prob, Tree(..), randomTree, runProb, uniform)
+import LazyPPL.Distributions.Counter (newCounter, readAndIncrement)
+import LazyPPL.Distributions.GP (gp, wiener)
+import LazyPPL.Distributions.IBP (Dish, newCustomer, newRestaurant)
+import LazyPPL.Distributions.Memoization (generalmemoize)
+import System.Environment (getArgs)
+import System.Exit (exitFailure)
+import System.Random (mkStdGen)
+import qualified KnownLimit
+
+-- A bind in Prob runs its first computation on the tree's first child.
+firstChild :: Tree -> Tree
+firstChild (Tree _ (t : _)) = t
+firstChild (Tree _ []) = error "a random tree has children"
+
+-- The answers of one IO action that runs once per key: it takes the next key
+-- and queries at it.
+queried :: (k -> IO a) -> [k] -> IO [a]
+queried query keys = do
+ remaining <- newIORef keys
+ replicateM (length keys) $ do
+ key <- atomicModifyIORef' remaining (\ks -> (drop 1 ks, head ks))
+ query key
+
+report :: String -> Bool -> String -> IO Bool
+report name ok detail = do
+ putStrLn ((if ok then "PASS " else "FAIL ") ++ name ++ ": " ++ detail)
+ pure ok
+
+-- Inside one Prob computation, the counter gives 0, 1, 2, ... in the order of
+-- the reads; read by an IO action after evaluate, it gives the same numbers.
+counterCheck :: IO [Bool]
+counterCheck = do
+ let tree = randomTree (mkStdGen 9)
+ n = 2000
+ inside <- mapM evaluate (runProb (do c <- newCounter; replicateM n (readAndIncrement c)) tree)
+ counter <- evaluate (runProb newCounter (firstChild tree))
+ outside <- queried (\k -> evaluate (runProb (readAndIncrement counter) (randomTree (mkStdGen k)))) [1 .. n]
+ sequence
+ [ report "counter read inside one Prob computation" (inside == [0 .. n - 1]) ("first reads " ++ show (take 5 inside))
+ , report "counter evaluated once, then read by an IO action" (outside == [0 .. n - 1]) ("first reads " ++ show (take 5 outside))
+ ]
+
+-- Six customers of one IBP restaurant take the first six rows of its matrix.
+restaurantCheck :: IO [Bool]
+restaurantCheck = do
+ let tree = randomTree (mkStdGen 9)
+ dishes = map show :: [Dish] -> [String]
+ inside <- mapM (evaluate . dishes) (runProb (do r <- newRestaurant 2; replicateM 6 (newCustomer r)) tree)
+ restaurant <- evaluate (runProb (newRestaurant 2) (firstChild tree))
+ outside <- queried (\k -> evaluate (dishes (runProb (newCustomer restaurant) (randomTree (mkStdGen k))))) [1 .. 6 :: Int]
+ sequence
+ [ report "restaurant customers inside one Prob computation differ" (length (nub inside) > 1) (show inside)
+ , report "restaurant evaluated once, then visited by an IO action" (outside == inside) (show outside)
+ ]
+
+-- A memoized, a Wiener and a Gaussian-process function queried at keys with
+-- a repeat. Inside one Prob computation, a repeated key gives its first answer
+-- and new keys give new answers; queried by an IO action after evaluate, the
+-- function gives the same answers.
+functionCheck :: String -> Prob (Double -> Double) -> Int -> IO [Bool]
+functionCheck name process seed = do
+ let tree = randomTree (mkStdGen seed)
+ keys = [1, 2, 1, 3, 2]
+ inside <- mapM evaluate (runProb (do f <- process; pure (map f keys)) tree)
+ f <- evaluate (runProb process (firstChild tree))
+ outside <- queried (evaluate . f) keys
+ sequence
+ [ report (name ++ " queried inside one Prob computation") (length (nub inside) == 3 && inside !! 2 == head inside)
+ ("at " ++ show keys ++ ": " ++ show inside)
+ , report (name ++ " evaluated once, then queried by an IO action") (outside == inside)
+ ("at " ++ show keys ++ ": " ++ show outside)
+ ]
+
+main :: IO ()
+main = do
+ args <- getArgs
+ results <- case args of
+ ["known-limit"] -> KnownLimit.checks
+ _ -> concat <$> sequence
+ [ counterCheck
+ , restaurantCheck
+ , functionCheck "generalmemoize" (generalmemoize (const uniform)) 10
+ , functionCheck "wiener" wiener 11
+ , functionCheck "white-noise gp" (gp (\x y -> if x == y then 1 else 0)) 12
+ ]
+ putStrLn (show (length (filter id results)) ++ "/" ++ show (length results) ++ " checks passed.")
+ unless (and results) exitFailure
diff --git a/test/KnownLimit.hs b/test/KnownLimit.hs
new file mode 100644
--- /dev/null
+++ b/test/KnownLimit.hs
@@ -0,0 +1,76 @@
+{-# OPTIONS_GHC -fno-cse -fno-full-laziness #-}
+-- The known limit of the values with hidden state that runProb returns: bound
+-- by let, or returned inside a function from a Prob computation, and then used
+-- by an IO action that runs more than once, such a value can lose its state.
+-- GHC treats an IO action's lambda as entered once (the state hack), so it may
+-- copy the binding into the action, and every run starts from new state. The
+-- options above are those of the concurrency test, where the copy happens.
+-- These checks fail while the limit holds; hidden-state-test runs them with
+-- the argument known-limit, and its default run does not.
+module KnownLimit (checks) where
+
+import Control.Concurrent (forkFinally, newEmptyMVar, putMVar, takeMVar)
+import Control.Exception (evaluate, throwIO)
+import Control.Monad (forM, replicateM)
+import Data.IORef (atomicModifyIORef', newIORef)
+import qualified Data.Set as Set
+import LazyPPL (randomTree, runProb)
+import LazyPPL.Distributions.Counter (newCounter, readAndIncrement)
+import LazyPPL.Distributions.GP (wiener)
+import System.Random (mkStdGen)
+
+-- Runs the action the given number of times in each of two threads at once.
+twoThreads :: Int -> IO a -> IO [a]
+twoThreads runs action = do
+ slots <- forM [1, 2 :: Int] $ \_ -> do
+ slot <- newEmptyMVar
+ _ <- forkFinally (replicateM runs (action >>= evaluate)) (putMVar slot)
+ pure slot
+ concat <$> mapM (\slot -> takeMVar slot >>= either throwIO pure) slots
+
+report :: String -> Bool -> String -> IO Bool
+report name ok detail = do
+ putStrLn ((if ok then "PASS " else "FAIL ") ++ "known limit, " ++ name ++ ": " ++ detail)
+ pure ok
+
+-- A counter bound by let, read 20000 times by each of two threads.
+counterLet :: IO Bool
+counterLet = do
+ let counter = runProb newCounter (randomTree (mkStdGen 9))
+ next k = runProb (readAndIncrement counter) (randomTree (mkStdGen k))
+ seeds <- newIORef (0 :: Int)
+ numbers <- twoThreads 20000 $ do
+ k <- atomicModifyIORef' seeds (\s -> (s + 1, s))
+ evaluate (next k)
+ let distinct = Set.size (Set.fromList numbers)
+ report "counter bound by let" (distinct == length numbers)
+ (show distinct ++ " distinct numbers from " ++ show (length numbers) ++ " reads")
+
+-- A function that reads a counter, returned from a Prob computation and bound by let.
+counterReturned :: IO Bool
+counterReturned = do
+ let next = runProb (do c <- newCounter
+ pure (\k -> runProb (readAndIncrement c) (randomTree (mkStdGen k))))
+ (randomTree (mkStdGen 9))
+ seeds <- newIORef (0 :: Int)
+ numbers <- twoThreads 20000 $ do
+ k <- atomicModifyIORef' seeds (\s -> (s + 1, s))
+ evaluate (next k)
+ let distinct = Set.size (Set.fromList numbers)
+ report "counter inside a function returned from Prob" (distinct == length numbers)
+ (show distinct ++ " distinct numbers from " ++ show (length numbers) ++ " reads")
+
+-- A Wiener function bound by let and queried at 1, then at 2, by two runs of one
+-- action. With one table, W(2) is drawn given W(1), and W(1) and W(2)/sqrt 2 differ.
+wienerLet :: IO Bool
+wienerLet = do
+ let f = runProb wiener (randomTree (mkStdGen 9))
+ times <- newIORef [1, 2 :: Double]
+ values <- replicateM 2 $ do
+ t <- atomicModifyIORef' times (\ts -> (drop 1 ts, head ts))
+ v <- evaluate (f t)
+ pure (v / sqrt t)
+ report "wiener bound by let" (values !! 0 /= values !! 1) ("W(1), W(2)/sqrt 2 = " ++ show values)
+
+checks :: IO [Bool]
+checks = sequence [counterLet, counterReturned, wienerLet]
diff --git a/test/Replay.hs b/test/Replay.hs
new file mode 100644
--- /dev/null
+++ b/test/Replay.hs
@@ -0,0 +1,123 @@
+module Main where
+
+import Control.Concurrent (forkFinally, newEmptyMVar, putMVar, takeMVar)
+import Control.Exception (evaluate, throwIO)
+import Control.Monad (forM, forM_, unless)
+import Data.List (nub)
+import Data.Word (Word64)
+import GHC.Float (castDoubleToWord64)
+import LazyPPL.Internal.History (History, appendMutation, labelAt, newHistory)
+import LazyPPL.Internal.Sampler (randomLabel)
+import System.Exit (exitFailure)
+import System.Random (StdGen, mkStdGen, split)
+import Test.QuickCheck
+
+-- Reference randomTree and mutateTree, as in LazyPPL 1.0.1, built without replay or the seed
+-- store.
+data RefTree = RefTree Double [RefTree]
+
+randomRefTree :: StdGen -> RefTree
+randomRefTree generator =
+ let (value, next) = randomLabel generator
+ in RefTree value (randomRefTrees next)
+
+randomRefTrees :: StdGen -> [RefTree]
+randomRefTrees generator =
+ let (left, right) = split generator
+ in randomRefTree left : randomRefTrees right
+
+mutateRefTree :: Double -> StdGen -> RefTree -> RefTree
+mutateRefTree rate generator (RefTree value children) =
+ let (decision, next) = randomLabel generator
+ (replacement, rest) = randomLabel next
+ in RefTree (if decision < rate then replacement else value)
+ (mutateRefTrees rate rest children)
+
+mutateRefTrees :: Double -> StdGen -> [RefTree] -> [RefTree]
+mutateRefTrees rate generator (tree:trees) =
+ let (left, right) = split generator
+ in mutateRefTree rate left tree : mutateRefTrees rate right trees
+mutateRefTrees _ _ [] = []
+
+referenceLabel :: [Int] -> RefTree -> Double
+referenceLabel [] (RefTree value _) = value
+referenceLabel (index:path) (RefTree _ children) =
+ referenceLabel path (children !! index)
+
+data ReplayCase = ReplayCase
+ { initialSeed :: Int
+ , proposalSeed :: Int
+ , prefixSteps :: Int
+ , laterSteps :: Int
+ , intermediateRate :: Double
+ , demandedPaths :: [[Int]]
+ } deriving Show
+
+instance Arbitrary ReplayCase where
+ arbitrary = do
+ initial <- chooseInt (minBound, maxBound)
+ proposals <- chooseInt (minBound, maxBound)
+ prefix <- oneof [chooseInt (0, 256), elements
+ [31,32,33,63,64,65,95,96,97,127,128,129,4095,4096,4097]]
+ later <- chooseInt (1, 96)
+ rate <- choose (0.0001, 0.9999)
+ paths <- listOf1 $ do
+ depth <- chooseInt (1, 4)
+ vectorOf depth (chooseInt (0, 5))
+ pure (ReplayCase initial proposals prefix later rate (nub ([] : take 6 paths)))
+ shrink input =
+ [input {prefixSteps=n} | n <- shrink (prefixSteps input), n >= 0] ++
+ [input {initialSeed=n} | n <- shrink (initialSeed input)] ++
+ [input {proposalSeed=n} | n <- shrink (proposalSeed input)] ++
+ [input {demandedPaths=[]} | not (null (demandedPaths input))]
+
+proposalGenerators :: StdGen -> [StdGen]
+proposalGenerators generator =
+ let (left, right) = split generator
+ in left : proposalGenerators right
+
+appendAll :: History -> Int -> [StdGen] -> IO ()
+appendAll history start generators =
+ forM_ (zip [start..] generators) $ \(index, generator) ->
+ appendMutation history index generator
+
+replayedLabels :: Double -> History -> Int -> [[Int]] -> IO [Word64]
+replayedLabels rate history count paths = do
+ -- Each path gets its own thread, so with -N2 labels replay concurrently.
+ slots <- forM paths $ \path -> do
+ slot <- newEmptyMVar
+ _ <- forkFinally
+ (evaluate (castDoubleToWord64 (labelAt rate history count (reverse path))))
+ (putMVar slot)
+ pure slot
+ answers <- mapM takeMVar slots
+ mapM (either throwIO pure) answers
+
+prop_replayPreservesDelayedLabels :: ReplayCase -> Property
+prop_replayPreservesDelayedLabels input =
+ classify (prefixSteps input >= 96) "at least 96 appended records" $
+ classify (prefixSteps input >= 4095) "near the 4096-record growth" $
+ classify (any ((> 1) . length) (demandedPaths input)) "nested paths" $
+ ioProperty $ do
+ let initial = mkStdGen (initialSeed input)
+ generators = proposalGenerators (mkStdGen (proposalSeed input))
+ (prefix, suffix) = splitAt (prefixSteps input) generators
+ paths = nub ([] : demandedPaths input)
+ results <- forM [0, intermediateRate input, 1] $ \rate -> do
+ history <- newHistory initial
+ appendAll history 0 prefix
+ -- Later records are appended before any label is demanded, and replay must ignore them.
+ appendAll history (prefixSteps input) (take (laterSteps input) suffix)
+ actual <- replayedLabels rate history (prefixSteps input) paths
+ let reference = foldl (\tree generator -> mutateRefTree rate generator tree)
+ (randomRefTree initial) prefix
+ expected = map (castDoubleToWord64 . (`referenceLabel` reference)) paths
+ pure $ counterexample ("rate=" ++ show rate ++ ", paths=" ++ show paths ++
+ ", actual=" ++ show actual ++ ", expected=" ++ show expected)
+ (actual === expected)
+ pure $ conjoin results
+
+main :: IO ()
+main = do
+ result <- quickCheckResult prop_replayPreservesDelayedLabels
+ unless (isSuccess result) exitFailure
diff --git a/test/Stream.hs b/test/Stream.hs
new file mode 100644
--- /dev/null
+++ b/test/Stream.hs
@@ -0,0 +1,50 @@
+module Main where
+
+import Control.Exception (evaluate)
+import Control.Monad (unless)
+import LazyPPL (Tree(..), mh, mhirreducible, runProb)
+import LazyPPL.Distributions (categorical)
+import System.Environment (getArgs)
+import System.Exit (exitFailure)
+
+check :: String -> Bool -> IO ()
+check name result = do
+ putStrLn $ (if result then "PASS " else "FAIL ") ++ name
+ unless result exitFailure
+
+categoricalRetention :: Bool -> IO ()
+categoricalRetention infinite = do
+ let n = 8000000
+ remaining = if infinite then 1 : repeat 0 else [1]
+ weights = replicate n 0 ++ remaining
+ label = let t = Tree 0.5 (repeat t) in t
+ selected <- evaluate (runProb (categorical weights) label)
+ check (if infinite then "categorical infinite stream fits the 256 MiB heap budget"
+ else "categorical finite stream fits the 256 MiB heap budget")
+ (selected == n)
+
+main :: IO ()
+main = do
+ args <- getArgs
+ case args of
+ ["categorical-finite"] -> categoricalRetention False
+ ["categorical-infinite"] -> categoricalRetention True
+ [] -> streamChecks >> categoricalRetention False >> categoricalRetention True
+ _ -> error "unknown stream test"
+
+streamChecks :: IO ()
+streamChecks = do
+ lazyDraws <- mh 0.1 (pure (repeat True))
+ check "infinite sample payload remains lazy"
+ (take 3 (fst (head lazyDraws)) == [True, True, True])
+ lazyIrreducibleDraws <- mhirreducible 0.1 1 (pure (repeat True))
+ check "irreducible infinite sample payload remains lazy"
+ (take 3 (fst (head lazyIrreducibleDraws)) == [True, True, True])
+ draws <- mhirreducible 0.2 1 (pure True)
+ result <- evaluate (fst (head (drop 4000000 draws)))
+ check "discarding a long chain preserves the constant model" result
+ partialDraws <- mh 0.2 (pure True)
+ -- Four million accepted mutations require about 61 MiB of packed seeds.
+ -- The RTS budget also allows collector growth and the sampler's live tree.
+ partialResult <- evaluate (fst (head (drop 4000000 partialDraws)))
+ check "partial-mutation seed records fit the 256 MiB heap budget" partialResult
diff --git a/test/StrictMemo.hs b/test/StrictMemo.hs
new file mode 100644
--- /dev/null
+++ b/test/StrictMemo.hs
@@ -0,0 +1,24 @@
+{-# OPTIONS_GHC -fdefer-type-errors -Wno-deferred-type-errors #-}
+-- This module defers type errors to run time, so that it compiles whether or
+-- not a strict monad has a MonadMemo instance.
+module StrictMemo (strictMemoChecks) where
+
+import Control.Exception (SomeException, TypeError(..), evaluate, fromException, try)
+import Data.Maybe (isJust)
+import LazyPPL.Distributions.Memoization (memoize)
+import System.Timeout (timeout)
+
+-- Memoization at Int builds an infinite trie inside the monad, which only a
+-- lazy monad such as Prob can return. A strict monad such as Maybe would
+-- never return the memo function, so it has no instance and the call below
+-- does not type-check.
+strictMemoChecks :: IO [Bool]
+strictMemoChecks = do
+ result <- try (timeout 200000 (evaluate (isJust (memoize (Just :: Int -> Maybe Int)))))
+ let (ok, outcome) = case result of
+ Left err | Just (TypeError _) <- fromException err -> (True, "rejected by the type checker")
+ | otherwise -> (False, "an exception: " ++ show (err :: SomeException))
+ Right Nothing -> (False, "no result within 0.2 s")
+ Right (Just _) -> (False, "a result")
+ putStrLn $ (if ok then "PASS" else "FAIL") ++ " memoize at Maybe Int gives " ++ outcome
+ pure [ok]
diff --git a/test/TestLogPdf.hs b/test/TestLogPdf.hs
new file mode 100644
--- /dev/null
+++ b/test/TestLogPdf.hs
@@ -0,0 +1,45 @@
+module Main where
+
+import LazyPPL.Distributions
+import System.Exit (exitFailure, exitSuccess)
+
+approxEq :: String -> Double -> Double -> Double -> IO Bool
+approxEq name expected actual tol = do
+ let ok = expected == actual || abs (expected - actual) < tol
+ if ok
+ then putStrLn $ " PASS: " ++ name
+ else putStrLn $ " FAIL: " ++ name ++ " expected=" ++ show expected ++ " actual=" ++ show actual
+ return ok
+
+main :: IO ()
+main = do
+ putStrLn "Testing log-PDF/PMF functions..."
+ results <- sequence
+ [ approxEq "normalLogPdf 0 1 0" (-0.9189385) (normalLogPdf 0 1 0) 1e-4
+ , approxEq "normalLogPdf 0 1 1" (-1.4189385) (normalLogPdf 0 1 1) 1e-4
+ , approxEq "exponentialLogPdf 1 1" (-1.0) (exponentialLogPdf 1 1) 1e-6
+ , approxEq "exponentialLogPdf 2 0.5" (log 2 - 1.0) (exponentialLogPdf 2 0.5) 1e-6
+ -- (k-1)*log(1) - 1/1 - 2*log(1) - logGamma(2) = -1
+ , approxEq "gammaLogPdf 2 1 1" (-1.0) (gammaLogPdf 2 1 1) 1e-6
+ , approxEq "betaLogPdf 1 1 0.5" 0.0 (betaLogPdf 1 1 0.5) 1e-6
+ -- B(2,2) = 1/6, pdf at 0.5 = 6*0.25 = 1.5
+ , approxEq "betaLogPdf 2 2 0.5" (log 1.5) (betaLogPdf 2 2 0.5) 1e-6
+ , approxEq "bernoulliLogPmf 0.3 True" (log 0.3) (bernoulliLogPmf 0.3 True) 1e-6
+ , approxEq "bernoulliLogPmf 0.3 False" (log 0.7) (bernoulliLogPmf 0.3 False) 1e-6
+ , approxEq "categoricalLogPmf [0.2,0.3,0.5] 2" (log 0.5) (categoricalLogPmf [0.2, 0.3, 0.5] 2) 1e-6
+ , approxEq "categoricalLogPmf OOB" (-1/0) (categoricalLogPmf [0.5, 0.5] 3) 1e-6 -- Outside the support: probability 0, log 0 = -Infinity.
+ , approxEq "uniformBoundedLogPdf 0 10 5" (-log 10) (uniformBoundedLogPdf 0 10 5) 1e-6
+ , approxEq "uniformBoundedLogPdf OOB" (-1/0) (uniformBoundedLogPdf 0 1 2) 1e-6 -- Outside the support: probability 0, log 0 = -Infinity.
+ -- Dirichlet(1,1,1) normalizer = Gamma(3)/(Gamma(1)^3) = 2
+ , approxEq "dirichletLogPdf [1,1,1] uniform" (log 2) (dirichletLogPdf [1,1,1] [1/3, 1/3, 1/3]) 1e-4
+ , approxEq "logNormalLogPdf 0 1 1" (-0.9189385) (logNormalLogPdf 0 1 1) 1e-4
+ , approxEq "logNormalLogPdf negative" (-1/0) (logNormalLogPdf 0 1 (-1)) 1e-6 -- Outside the support: probability 0, log 0 = -Infinity.
+ -- C(10,5) = 252
+ , approxEq "binomialLogPmf 10 0.5 5" (log 252 - 10 * log 2) (binomialLogPmf 10 0.5 5) 1e-4
+ , approxEq "binomialLogPmf OOB" (-1/0) (binomialLogPmf 5 0.3 6) 1e-6 -- Outside the support: probability 0, log 0 = -Infinity.
+ , approxEq "poissonLogPmf 3 2" (2 * log 3 - 3 - log 2) (poissonLogPmf 3 2) 1e-6
+ ]
+ let nPass = length (filter id results)
+ nTotal = length results
+ putStrLn $ show nPass ++ "/" ++ show nTotal ++ " tests passed."
+ if nPass == nTotal then exitSuccess else exitFailure
diff --git a/test/reference/beta-laws.tsv b/test/reference/beta-laws.tsv
new file mode 100644
--- /dev/null
+++ b/test/reference/beta-laws.tsv
@@ -0,0 +1,4274 @@
+# Written by test/reference/reference.py with mpmath 1.4.1.
+# a b x P(X <= x), nearest double
+2.5 3.5 0.024735511935950617 0.001
+2.5 3.5 0.03283142434080519 0.0020000000000000005
+2.5 3.5 0.03878049240662356 0.0030000000000000005
+2.5 3.5 0.04366604159651973 0.004
+2.5 3.5 0.04789128932392727 0.005000000000000001
+2.5 3.5 0.05165760451749903 0.005999999999999999
+2.5 3.5 0.05508220932156212 0.007
+2.5 3.5 0.05824023346830339 0.008000000000000002
+2.5 3.5 0.06118314256625607 0.009
+2.5 3.5 0.06394796069062224 0.01
+2.5 3.5 0.06656234329733389 0.011
+2.5 3.5 0.06904757312926366 0.011999999999999999
+2.5 3.5 0.07142043063430868 0.013
+2.5 3.5 0.07369441515898603 0.014
+2.5 3.5 0.07588057219558705 0.015000000000000001
+2.5 3.5 0.07798807135447577 0.016
+2.5 3.5 0.08002462093794613 0.017
+2.5 3.5 0.08199677212798966 0.018000000000000002
+2.5 3.5 0.08391014663077564 0.019000000000000003
+2.5 3.5 0.08576961002160641 0.019999999999999997
+2.5 3.5 0.08757940578842681 0.020999999999999998
+2.5 3.5 0.08934326041719004 0.022
+2.5 3.5 0.09106446679696091 0.023000000000000003
+2.5 3.5 0.09274595115872164 0.024
+2.5 3.5 0.09439032734434406 0.025
+2.5 3.5 0.09599994121101994 0.025999999999999995
+2.5 3.5 0.09757690727199335 0.026999999999999996
+2.5 3.5 0.09912313916627749 0.028000000000000004
+2.5 3.5 0.10064037517845022 0.029000000000000005
+2.5 3.5 0.10213019975448094 0.03
+2.5 3.5 0.1035940617534504 0.030999999999999996
+2.5 3.5 0.10503329001899721 0.031999999999999994
+2.5 3.5 0.10644910673502296 0.033
+2.5 3.5 0.10784263893812121 0.034
+2.5 3.5 0.10921492848753295 0.035
+2.5 3.5 0.11056694073719951 0.03599999999999999
+2.5 3.5 0.11189957211002574 0.037
+2.5 3.5 0.11321365673906711 0.038
+2.5 3.5 0.11450997231197947 0.039
+2.5 3.5 0.11578924523218148 0.04
+2.5 3.5 0.11705215519160758 0.041
+2.5 3.5 0.11829933923477379 0.04200000000000001
+2.5 3.5 0.11953139538144572 0.043
+2.5 3.5 0.120748885864947 0.04399999999999999
+2.5 3.5 0.12195234003465266 0.045
+2.5 3.5 0.12314225696414395 0.04599999999999999
+2.5 3.5 0.124319107800592 0.047
+2.5 3.5 0.12548333788597762 0.048
+2.5 3.5 0.12663536867657577 0.049000000000000016
+2.5 3.5 0.12777559948359626 0.05
+2.5 3.5 0.1289044090548724 0.05100000000000001
+2.5 3.5 0.13002215701493 0.05200000000000001
+2.5 3.5 0.13112918517858535 0.05299999999999999
+2.5 3.5 0.1322258187513454 0.054000000000000006
+2.5 3.5 0.13331236742827104 0.055
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diff --git a/test/reference/beta-quantiles.tsv b/test/reference/beta-quantiles.tsv
new file mode 100644
--- /dev/null
+++ b/test/reference/beta-quantiles.tsv
@@ -0,0 +1,7417 @@
+# Written by test/reference/reference.py with mpmath 1.4.1.
+# a b u x low or reject x high 1-x low 1-x high smaller side smaller side exact linearised half-width (informational)
+1e-10 1e-10 1.1102230246251565e-16 reject
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+1e-10 1.0 1e-300 0.0 5e-324 0.9999999999999999 1.0 x 1.000251947027571786e-3000000000000 0.0
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+1e-10 1.0 1e-20 0.0 5e-324 0.9999999999999999 1.0 x 1.0000162293112971264e-200000000000 0.0
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+1e-10 1.0 0.75 0.0 5e-324 0.9999999999999999 1.0 x 8.2603892758531428655e-1249387367 0.0
+1e-10 1.0 0.999 0.0 5e-324 0.9999999999999999 1.0 x 1.8189597254773064143e-4345118 0.0
+1e-10 1.0 0.99999999 3.720071874728862e-44 3.7200726187433216e-44 0.9999999999999999 1.0 x 3.7200722467360731282e-44 3.720072302629264e-51
+1e-10 1.0 0.9999999990686774 9.022303741381665e-05 9.022303909435169e-05 0.9999097769609057 0.9999097769625861 x 0.000090223038254084164068 8.402675235468048e-13
+1e-10 1.0 0.9999999999999999 0.9999988897775907 0.9999988897775928 1.1102224072175808e-06 1.110222409438024e-06 y 1.1102224083278024335e-6 1.1102217920306765e-15
+1e-10 2.0 1.1102230246251565e-16 reject
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+0.1 1.0 1.1102230246251565e-16 2.8451311708896455e-160 2.845131227792269e-160 0.9999999999999999 1.0 x 2.8451311993409571996e-160 2.845131199340957e-168
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+0.1 2.0 1e-300 reject
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+0.1 2.0 1e-08 3.8554328557410294e-81 3.8554329328496865e-81 0.9999999999999999 1.0 x 3.8554328942953577136e-81 3.8554328942953576e-89
+0.1 2.0 0.001 3.8554328557410046e-31 3.855432932849662e-31 0.9999999999999999 1.0 x 3.8554328942953330696e-31 3.8554328942953326e-39
+0.1 2.0 0.25 3.676828522154982e-07 3.676828595691577e-07 0.9999996323171405 0.9999996323171477 x 3.6768285589232794849e-7 3.676829787929936e-15
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+0.1 30.0 1.1102230246251565e-16 5.846972297799083e-162 5.84697241473853e-162 0.9999999999999999 1.0 x 5.8469723562688064672e-162 5.8469723562688064e-170
+0.1 30.0 1e-300 reject
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+30.0 0.5 0.99999999 0.9999999999999999 1.0 2.6399006523494505e-18 2.639900662909053e-18 y 2.6399006576292516363e-18 5.279801315258503e-27
+30.0 0.5 0.9999999990686774 0.9999999999999999 1.0 2.2897487949276664e-20 2.2897488040866613e-20 y 2.2897487995071639649e-20 4.579497599014328e-29
+30.0 0.5 0.9999999999999999 0.9999999999999999 1.0 3.2539287459025934e-34 3.253928758918308e-34 y 3.2539287524104508462e-34 6.5078575048209015e-43
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+30.0 1.0 1e-300 9.999999999666667e-11 1.0000000000333332e-10 0.9999999999 0.9999999999000001 x 1.0000000000000000008e-10 3.3333333333333333e-21
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+30.0 1.0 0.99999999 0.9999999996666666 0.9999999996666668 3.3333333628603093e-10 3.3333333695269756e-10 y 3.3333333661936422964e-10 3.333333382304754e-19
+30.0 1.0 0.9999999990686774 0.9999999999689558 0.9999999999689559 3.104408580344603e-11 3.1044085865534195e-11 y 3.1044085834490111864e-11 3.104408584846427e-20
+30.0 1.0 0.9999999999999999 0.9999999999999999 1.0 3.7007434117164455e-18 3.700743419117931e-18 y 3.7007434154171886667e-18 3.7007434154171886e-27
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+30.0 2.0 1e-100 reject
+30.0 2.0 1e-20 reject
+30.0 2.0 1e-08 0.4931877300637458 0.49318773009765704 0.506812269902343 0.5068122699362542 x 0.49318773008070140625 1.6955644181408843e-11
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+30.0 2.0 0.99999999 0.99999536240313 0.9999953624031346 4.637596865415606e-06 4.63759687005341e-06 y 4.6375968677345078502e-6 2.318902389696824e-15
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+30.0 30.0 1e-100 reject
+30.0 30.0 1e-20 reject
+30.0 30.0 1e-08 0.17950480697456087 0.17950480698954638 0.8204951930104537 0.820495193025439 x 0.17950480698205363332 7.492770695530568e-12
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+30.0 100.0 1e-20 reject
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+53.5 0.001 0.5 0.9999999999999999 1.0 9.894479075421623e-304 9.894498864399561e-304 y 9.8944889699056490369e-304 9.89448896990563e-310
+53.5 0.001 0.75 0.9999999999999999 1.0 0.0 5e-324 y 9.2341665792944660265e-605 0.0
+53.5 0.001 0.999 0.9999999999999999 1.0 0.0 5e-324 y 1.0602030094974274926e-3002 0.0
+53.5 0.001 0.99999999 0.9999999999999999 1.0 0.0 5e-324 y 1.0602083367750163834e-8002 0.0
+53.5 0.001 0.9999999990686774 0.9999999999999999 1.0 0.0 5e-324 y 1.3351163465748192404e-9033 0.0
+53.5 0.001 0.9999999999999999 0.9999999999999999 1.0 0.0 5e-324 y 2.7265837476250821523e-15957 0.0
+53.5 0.0018614780417212304 1.1102230246251565e-16 0.599583240690012 0.5995832407118423 0.4004167592881577 0.400416759309988 y 0.40041675929907283182 1.0915191878668792e-11
+53.5 0.0018614780417212304 1e-300 reject
+53.5 0.0018614780417212304 1e-100 reject
+53.5 0.0018614780417212304 1e-20 reject
+53.5 0.0018614780417212304 1e-08 0.8316736893533214 0.831673689382 0.16832631061799996 0.1683263106466786 y 0.16832631063233929582 1.433933230350004e-11
+53.5 0.0018614780417212304 0.001 0.9902442038437484 0.9902442038612884 0.00975579613871158 0.009755796156251651 y 0.0097557961474816148987 8.770036240966037e-12
+53.5 0.0018614780417212304 0.25 0.9999999999999999 1.0 8.084702100472376e-70 8.084704995914731e-70 y 8.0847035481934237925e-70 1.447721177650465e-76
+53.5 0.0018614780417212304 0.5 0.9999999999999999 1.0 2.0422810876077845e-164 2.0422832818664742e-164 y 2.0422821847368352656e-164 1.0971293450490681e-170
+53.5 0.0018614780417212304 0.75 0.9999999999999999 1.0 0.0 5e-324 y 3.931322835800259214e-326 0.0
+53.5 0.0018614780417212304 0.999 0.9999999999999999 1.0 0.0 5e-324 y 2.5300933192333602366e-1614 0.0
+53.5 0.0018614780417212304 0.99999999 0.9999999999999999 1.0 0.0 5e-324 y 2.3202990593915870295e-4300 0.0
+53.5 0.0018614780417212304 0.9999999990686774 0.9999999999999999 1.0 0.0 5e-324 y 3.6172755502649325386e-4854 0.0
+53.5 0.0018614780417212304 0.9999999999999999 0.9999999999999999 1.0 0.0 5e-324 y 1.2592433791227099959e-8573 0.0
+53.5 0.01 1.1102230246251565e-16 0.5811751897065983 0.5811751897278021 0.4188248102721978 0.41882481029340174 y 0.41882481028279976456 1.0601976475723258e-11
+53.5 0.01 1e-300 reject
+53.5 0.01 1e-100 reject
+53.5 0.01 1e-20 reject
+53.5 0.01 1e-08 0.8074381558660961 0.8074381558942672 0.19256184410573282 0.19256184413390387 y 0.19256184411981835825 1.4085538120016917e-11
+53.5 0.01 0.001 0.9719318919051082 0.9719318919298917 0.028068108070108188 0.028068108094891835 y 0.028068108082500011114 1.2391825333256663e-11
+53.5 0.01 0.25 0.9999999999999966 0.9999999999999967 3.4251225256894702e-15 3.4251227540309793e-15 y 3.4251226398602229894e-15 1.1417075466202776e-22
+53.5 0.01 0.5 0.9999999999999999 1.0 8.424617220247732e-33 8.424618905171343e-33 y 8.4246180627094950585e-33 8.424618062709495e-40
+53.5 0.01 0.75 0.9999999999999999 1.0 6.645851166505159e-63 6.645852495675524e-63 y 6.6458518310903086612e-63 6.645851831090308e-70
+53.5 0.01 0.999 0.9999999999999999 1.0 1.06794710759412e-302 1.0679473211835628e-302 y 1.067947214388836126e-302 1.067947214388836e-309
+53.5 0.01 0.99999999 0.9999999999999999 1.0 0.0 5e-324 y 1.0679477510066674446e-802 0.0
+53.5 0.01 0.9999999990686774 0.9999999999999999 1.0 0.0 5e-324 y 8.6808608616880660435e-906 0.0
+53.5 0.01 0.9999999999999999 0.9999999999999999 1.0 0.0 5e-324 y 3.7116991562614383428e-1598 0.0
+53.5 0.1 1.1102230246251565e-16 0.5538808462549671 0.5538808462752653 0.44611915372473465 0.44611915374503286 y 0.44611915373488375495 1.0149141073707546e-11
+53.5 0.1 1e-300 reject
+53.5 0.1 1e-100 reject
+53.5 0.1 1e-20 reject
+53.5 0.1 1e-08 0.7717249156148019 0.7717249156422006 0.22827508435779936 0.22827508438519814 y 0.22827508437149874603 1.3699406640577555e-11
+53.5 0.1 0.001 0.938566078552395 0.9385660785813726 0.061433921418627274 0.06143392144760498 y 0.061433921433116129351 1.4488855402635132e-11
+53.5 0.1 0.25 0.999334701632964 0.9993347016375425 0.0006652983624574564 0.0006652983670360328 y 0.000665298364746744654 2.2892882270266745e-12
+53.5 0.1 0.5 0.9999888147202596 0.9999888147204833 1.1185279516650903e-05 1.1185279740475952e-05 y 0.000011185279628563426737 1.1191252515635341e-13
+53.5 0.1 0.75 0.9999999890826986 0.9999999890826987 1.091730126399335e-08 1.091730148233949e-08 y 1.0917301373166419883e-8 1.0917307061660856e-16
+53.5 0.1 0.999 0.9999999999999999 1.0 1.1447614125378544e-32 1.1447614354330828e-32 y 1.1447614239854685185e-32 1.1447614239854685e-40
+53.5 0.1 0.99999999 0.9999999999999999 1.0 1.144761470059358e-82 1.1447614929545876e-82 y 1.1447614815069727999e-82 1.1447614815069728e-90
+53.5 0.1 0.9999999990686774 0.9999999999999999 1.0 5.619740769614569e-93 5.619740882009385e-93 y 5.6197408258119770395e-93 5.6197408258119765e-101
+53.5 0.1 0.9999999999999999 0.9999999999999999 1.0 3.256996410613033e-162 3.256996475752961e-162 y 3.256996443182996637e-162 3.2569964431829965e-170
+53.5 0.5 1.1102230246251565e-16 0.5244029832947155 0.5244029833141265 0.47559701668587345 0.47559701670528454 y 0.47559701669557900075 9.705590384996614e-12
+53.5 0.5 1e-300 reject
+53.5 0.5 1e-100 reject
+53.5 0.5 1e-20 reject
+53.5 0.5 1e-08 0.7346742890600751 0.7346742890868885 0.26532571091311147 0.2653257109399249 y 0.26532571092651820782 1.3406741415146456e-11
+53.5 0.5 0.001 0.9033347144223006 0.9033347144537072 0.09666528554629279 0.0966652855776994 y 0.096665285561996101258 1.5703311610341228e-11
+53.5 0.5 0.25 0.9876516844342629 0.9876516844601739 0.012348315539826026 0.012348315565737171 y 0.012348315552781598994 1.2955573585944995e-11
+53.5 0.5 0.5 0.9957374612165079 0.9957374612363523 0.004262538763647599 0.004262538783492204 y 0.0042625387735699013094 9.922302951746893e-12
+53.5 0.5 0.75 0.9990471219976768 0.9990471220016179 0.0009528779983820405 0.0009528780023233251 y 0.00095287800035268281466 1.9706423459097384e-12
+53.5 0.5 0.999 0.9999999852508936 0.9999999852508937 1.4749106330784959e-08 1.4749106389781412e-08 y 1.4749106360283185568e-8 2.949822794810091e-17
+53.5 0.5 0.99999999 0.9999999999999999 1.0 1.4749098865244179e-18 1.474909892424057e-18 y 1.4749098894742374586e-18 2.9498197789484752e-27
+53.5 0.5 0.9999999990686774 0.9999999999999999 1.0 1.279280389695946e-20 1.2792803948130673e-20 y 1.2792803922545066542e-20 2.5585607845090134e-29
+53.5 0.5 0.9999999999999999 0.9999999999999999 1.0 1.817966775797609e-34 1.817966783069476e-34 y 1.8179667794335424753e-34 3.635933558867085e-43
+53.5 1.0 1.1102230246251565e-16 0.5032495193546443 0.5032495193734573 0.4967504806265427 0.4967504806453557 y 0.49675048063594920919 9.406533072225248e-12
+53.5 1.0 1e-300 2.4690129310153474e-06 2.4690129311076464e-06 0.9999975309870689 0.999997530987069 x 2.4690129310614968768e-6 4.6149774412364427e-17
+53.5 1.0 1e-100 0.013515780232019674 0.013515780232524935 0.9864842197674751 0.9864842197679803 x 0.013515780232272305095 2.5263140621069727e-13
+53.5 1.0 1e-20 0.42283236698088517 0.4228323669966919 0.5771676330033081 0.5771676330191148 x 0.42283236698878852839 7.903408728762403e-12
+53.5 1.0 1e-08 0.7087079258195772 0.7087079258460708 0.29129207415392916 0.29129207418042286 y 0.29129207416717598448 1.3246877118370542e-11
+53.5 1.0 0.001 0.8788712009318539 0.8788712009647088 0.12112879903529114 0.12112879906814612 y 0.12112879905171863164 1.6427499083145448e-11
+53.5 1.0 0.25 0.9744207918469638 0.9744207918833907 0.02557920811660922 0.025579208153036166 y 0.025579208134822693828 1.82134727451435e-11
+53.5 1.0 0.5 0.9871275458767983 0.9871275459137001 0.012872454086299823 0.012872454123201786 y 0.012872454104750804836 1.845098216626634e-11
+53.5 1.0 0.75 0.9946371964441367 0.9946371964565308 0.005362803543469117 0.0053628035558633495 y 0.0053628035496662333529 6.197116488787126e-12
+53.5 1.0 0.999 0.9999812992340282 0.9999812992340655 1.8700765934431904e-05 1.87007659718518e-05 y 0.000018700765953141852741 1.870994918720679e-14
+53.5 1.0 0.99999999 0.9999999998130841 0.9999999998130842 1.8691588951986943e-10 1.869158898937012e-10 y 1.8691588970678531086e-10 1.86915890623896e-19
+53.5 1.0 0.9999999990686774 0.999999999982592 0.9999999999825921 1.7407898581490292e-11 1.740789861630609e-11 y 1.7407898598898190662e-11 1.7407898606852857e-20
+53.5 1.0 0.9999999999999999 0.9999999999999999 1.0 2.0751832215232407e-18 2.0751832256736066e-18 y 2.0751832235984235531e-18 2.0751832235984237e-27
+53.5 2.0 1.1102230246251565e-16 0.4724837629019985 0.47248376291995164 0.5275162370800484 0.5275162370980014 x 0.47248376291097507631 8.976612385040642e-12
+53.5 2.0 1e-300 reject
+53.5 2.0 1e-100 reject
+53.5 2.0 1e-20 reject
+53.5 2.0 1e-08 0.6710238846027798 0.6710238846288036 0.3289761153711964 0.3289761153972202 y 0.32897611538420830339 1.3011922209457276e-11
+53.5 2.0 0.001 0.8428055361981485 0.8428055362327547 0.15719446376724533 0.1571944638018516 y 0.15719446378454844781 1.7303144433343308e-11
+53.5 2.0 0.25 0.9513553196825173 0.9513553197308442 0.04864468026915578 0.048644680317482786 y 0.048644680293319280807 2.4163505804426487e-11
+53.5 2.0 0.5 0.969395519302386 0.9693955193596868 0.03060448064031317 0.030604480697614112 y 0.030604480668963642013 2.8650472210529787e-11
+53.5 2.0 0.75 0.9823550614473714 0.9823550614721173 0.017644938527882705 0.017644938552628647 y 0.017644938540255675518 1.2372971132421411e-11
+53.5 2.0 0.999 0.9991595386889415 0.9991595386897943 0.0008404613102056225 0.0008404613110585908 y 0.00084046131063210669533 4.2648420511215397e-13
+53.5 2.0 0.99999999 0.9999973808536703 0.9999973808536728 2.619146327126333e-06 2.619146329745599e-06 y 2.6191463284359660417e-6 1.3096331907649077e-15
+53.5 2.0 0.9999999990686774 0.99999920072543 0.9999992007254307 7.992745692318087e-07 7.992745700310943e-07 y 7.99274569631451459e-7 3.9964287472629687e-16
+53.5 2.0 0.9999999999999999 0.9999999997240406 0.9999999997240407 2.7595926049325386e-10 2.759592607692131e-10 y 2.7595926063123347591e-10 1.3797963098196e-19
+53.5 30.0 1.1102230246251565e-16 0.2210246357301058 0.22102463573982217 0.7789753642601779 0.7789753642698941 x 0.22102463573496399078 4.858203217852815e-12
+53.5 30.0 1e-300 reject
+53.5 30.0 1e-100 reject
+53.5 30.0 1e-20 reject
+53.5 30.0 1e-08 0.3393947206948843 0.3393947207122256 0.6606052792877745 0.6606052793051157 x 0.33939472070355494608 8.67068282314933e-12
+53.5 30.0 0.001 0.4731058386124729 0.47310583864514144 0.5268941613548586 0.5268941613875271 x 0.47310583862880717126 1.6334278056922907e-11
+53.5 30.0 0.25 0.6058835921010067 0.6058835921858887 0.3941164078141113 0.39411640789899327 y 0.39411640785655229493 4.2441017726909724e-11
+53.5 30.0 0.5 0.6418471408424911 0.6418471409743912 0.3581528590256088 0.35815285915750894 y 0.35815285909155887678 6.595010235340529e-11
+53.5 30.0 0.75 0.676774238343911 0.6767742384239654 0.3232257615760345 0.32322576165608896 y 0.32322576161606174719 4.002725083497947e-11
+53.5 30.0 0.999 0.7895452184247465 0.7895452184496076 0.21045478155039243 0.21045478157525355 y 0.2104547815628229892 1.2430576327007878e-11
+53.5 30.0 0.99999999 0.8802080088994803 0.8802080089098145 0.11979199109018544 0.11979199110051982 y 0.11979199109535263231 5.167190775359722e-12
+53.5 30.0 0.9999999990686774 0.8916932672193236 0.8916932672283849 0.10830673277161497 0.10830673278067642 y 0.10830673277614569595 4.530729569624889e-12
+53.5 30.0 0.9999999999999999 0.9419360361928706 0.9419360361971898 0.05806396380281017 0.05806396380712944 y 0.058063963804969804302 2.159641471899329e-12
+53.5 53.5 1.1102230246251565e-16 0.15764522217096005 0.157645222178137 0.8423547778218631 0.8423547778290399 x 0.1576452221745485306 3.5884835813124953e-12
+53.5 53.5 1e-300 reject
+53.5 53.5 1e-100 reject
+53.5 53.5 1e-20 reject
+53.5 53.5 1e-08 0.24702812775929717 0.24702812777269983 0.7529718722273002 0.7529718722407028 x 0.24702812776599849333 6.701349938055254e-12
+53.5 53.5 0.001 0.3535754048419393 0.35357540486884015 0.6464245951311599 0.6464245951580607 x 0.35357540485538970587 1.3450468963557972e-11
+53.5 53.5 0.25 0.4673559393476692 0.4673559394236578 0.5326440605763423 0.5326440606523307 x 0.46735593938566350279 3.799428887298588e-11
+53.5 53.5 0.5 0.4999999999392771 0.5000000000607229 0.4999999999392771 0.5000000000607229 x 0.5 6.072293377818844e-11
+53.5 53.5 0.75 0.5326440605763423 0.5326440606523307 0.4673559393476692 0.4673559394236578 y 0.46735593938566350279 3.799428887298588e-11
+53.5 53.5 0.999 0.6464245951311599 0.6464245951580607 0.3535754048419393 0.35357540486884015 y 0.35357540485538971753 1.3450468963557974e-11
+53.5 53.5 0.99999999 0.7529718721936275 0.7529718722070301 0.24702812779296981 0.2470281278063725 y 0.24702812779967116337 6.701349939504163e-12
+53.5 53.5 0.9999999990686774 0.7681146956176675 0.7681146956298196 0.2318853043701804 0.23188530438233246 y 0.23188530437625642597 6.076032019447075e-12
+53.5 53.5 0.9999999999999999 0.8423547778218631 0.8423547778290399 0.15764522217096005 0.157645222178137 y 0.1576452221745485306 3.5884835813124953e-12
+53.5 100.0 1.1102230246251565e-16 0.10127343171199225 0.10127343171674402 0.898726568283256 0.8987265682880077 x 0.1012734317143681382 2.375885355475851e-12
+53.5 100.0 1e-300 reject
+53.5 100.0 1e-100 reject
+53.5 100.0 1e-20 reject
+53.5 100.0 1e-08 0.1615921232722637 0.16159212328148834 0.8384078767185117 0.8384078767277363 x 0.16159212327687601525 4.612340890221421e-12
+53.5 100.0 0.001 0.2368560202960451 0.23685602031557124 0.7631439796844288 0.7631439797039549 x 0.23685602030580817345 9.763075277125438e-12
+53.5 100.0 0.25 0.32222464378382204 0.3222246438428699 0.6777753561571301 0.6777753562161779 x 0.32222464381334599304 2.952397084446718e-11
+53.5 100.0 0.5 0.3478747641493194 0.34787476424583275 0.6521252357541673 0.6521252358506806 x 0.34787476419757608928 4.825670280659375e-11
+53.5 100.0 0.75 0.37412793205804723 0.3741279321199061 0.625872067880094 0.6258720679419527 x 0.37412793208897664806 3.092946863846346e-11
+53.5 100.0 0.999 0.47132488211422363 0.47132488213843793 0.5286751178615621 0.5286751178857764 x 0.47132488212633079335 1.2107190183519867e-11
+53.5 100.0 0.99999999 0.5734123836840002 0.5734123836977644 0.4265876163022356 0.42658761631599984 y 0.42658761630911771194 6.882142522126367e-12
+53.5 100.0 0.9999999990686774 0.589146391772982 0.5891463917857594 0.4108536082142405 0.410853608227018 y 0.41085360822062925355 6.388786869562954e-12
+53.5 100.0 0.9999999999999999 0.6727703720822827 0.672770372091014 0.327229627908986 0.3272296279177173 y 0.3272296279133516761 4.3656795107992005e-12
+53.5 1000.0 1.1102230246251565e-16 0.012929661108535247 0.012929661109170152 0.98707033889083 0.9870703388914647 x 0.012929661108852699476 3.1745325625853387e-13
+53.5 1000.0 1e-300 reject
+53.5 1000.0 1e-100 reject
+53.5 1000.0 1e-20 reject
+53.5 1000.0 1e-08 0.021227250538911564 0.021227250540218543 0.9787727494597815 0.9787727494610884 x 0.021227250539565053539 6.534899529314831e-13
+53.5 1000.0 0.001 0.03231874667300908 0.03231874667600424 0.9676812533239958 0.9676812533269908 x 0.03231874667450665962 1.4975833765137762e-12
+53.5 1000.0 0.25 0.04607854210989907 0.04607854211991126 0.9539214578800888 0.9539214578901009 x 0.046078542114905165018 5.0060978968062196e-12
+53.5 1000.0 0.5 0.050498996738150205 0.05049899675505738 0.9495010032449427 0.9495010032618497 x 0.050498996746603795184 8.453590694323467e-12
+53.5 1000.0 0.75 0.055178257235202866 0.05517825724642227 0.9448217427535778 0.9448217427647971 x 0.055178257240812569519 5.6097064914670285e-12
+53.5 1000.0 0.999 0.074089014431756 0.07408901443682163 0.9259109855631784 0.9259109855682439 x 0.074089014434288812723 2.5328243510834727e-12
+53.5 1000.0 0.99999999 0.09743254355896115 0.09743254356241436 0.9025674564375857 0.9025674564410388 x 0.097432543560687759572 1.7266146241759731e-12
+53.5 1000.0 0.9999999990686774 0.10144308230038739 0.10144308230369681 0.8985569176963032 0.8985569176996125 x 0.10144308230204209893 1.6547176320027799e-12
+53.5 1000.0 0.9999999999999999 0.1252463445079073 0.1252463445106504 0.8747536554893497 0.8747536554920926 x 0.12524634450927884054 1.3715526522311455e-12
+53.5 10000.0 1.1102230246251565e-16 reject
+53.5 10000.0 1e-300 reject
+53.5 10000.0 1e-100 reject
+53.5 10000.0 1e-20 reject
+53.5 10000.0 1e-08 reject
+53.5 10000.0 0.001 reject
+53.5 10000.0 0.25 reject
+53.5 10000.0 0.5 reject
+53.5 10000.0 0.75 reject
+53.5 10000.0 0.999 reject
+53.5 10000.0 0.99999999 reject
+53.5 10000.0 0.9999999990686774 reject
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diff --git a/test/reference/dirichlet-sticks.tsv b/test/reference/dirichlet-sticks.tsv
new file mode 100644
--- /dev/null
+++ b/test/reference/dirichlet-sticks.tsv
@@ -0,0 +1,57 @@
+# Written by test/reference/reference.py with mpmath 1.4.1.
+# alphas u coordinate 1 low coordinate 1 high ...
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diff --git a/test/reference/log-densities.tsv b/test/reference/log-densities.tsv
new file mode 100644
--- /dev/null
+++ b/test/reference/log-densities.tsv
@@ -0,0 +1,3075 @@
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diff --git a/test/reference/reference.py b/test/reference/reference.py
new file mode 100644
--- /dev/null
+++ b/test/reference/reference.py
@@ -0,0 +1,646 @@
+"""Reference values for the Beta and Dirichlet samplers and the log densities.
+
+Run from the package root:
+
+ uv run --with mpmath==1.4.1 python -I test/reference/reference.py [--jobs N]
+
+It writes the tables beside this file, which test/Api.hs reads. Every value is
+computed by mpmath at high precision from the definition of the distribution:
+the incomplete Beta function (mpmath's betainc, or mpmath's quad of the
+density), mpmath's loggamma and logarithms. None comes from the Haskell code.
+
+Tables:
+
+ beta-quantiles.tsv For Beta(a,b) and a label u, the exact u-quantile x and
+ 1 - x, and for each the interval of doubles that the samplers may return:
+ every double whose distribution function differs from u by at most
+ EPS * min(u, 1 - u), joined with the two doubles next to the exact value.
+ A row marked "reject" names an input rejected by both samplers; it has no
+ quantile intervals.
+ dirichlet-sticks.tsv dirichlet as at the tree whose every label is u: the
+ exact stick-breaking coordinates, each with a relative tolerance built
+ from the same intervals.
+ beta-laws.tsv Distribution functions at grids of points, for law checks of
+ two general Beta shapes and the four nonparametric stick laws.
+ log-densities.tsv Log densities and masses, with the exact value rounded to
+ the nearest double (or an infinity where it leaves the double range).
+"""
+
+import argparse
+import math
+import os
+import struct
+import sys
+from multiprocessing import Pool
+
+import mpmath as mp
+
+# mpmath 1.4.1 is the latest stable release (PyPI, 15 March 2026). The tables
+# are reproducible only for the version that wrote them.
+MPMATH_VERSION = "1.4.1"
+if mp.__version__ != MPMATH_VERSION:
+ sys.exit(f"reference.py needs mpmath {MPMATH_VERSION}, found {mp.__version__}")
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+
+# Relative tolerance on the probability of a returned quantile.
+EPS = "1e-9"
+
+MAXD = sys.float_info.max
+
+
+# Doubles and exact values -------------------------------------------------
+
+def exact(v):
+ """The exact value of a double or an integer, as an mpf."""
+ return mp.mpf(v)
+
+
+def down(v):
+ """The largest double at most v (v an mpf in the double range)."""
+ if v <= 0:
+ return 0.0 if v == 0 else -math.inf
+ f = float(mp.nstr(v, 25, strip_zeros=False))
+ while f > 0 and exact(f) > v:
+ f = math.nextafter(f, -math.inf)
+ while math.nextafter(f, math.inf) <= MAXD and exact(math.nextafter(f, math.inf)) <= v:
+ f = math.nextafter(f, math.inf)
+ return f
+
+
+def up(v):
+ """The smallest double at least v."""
+ f = down(v)
+ return f if exact(f) == v else math.nextafter(f, math.inf)
+
+
+def nearest(v):
+ """The double nearest to v, ties to even, with infinities beyond the range:
+ from MAXD plus half its spacing 2^970 up, IEEE rounding gives infinity."""
+ if abs(v) >= exact(MAXD) + mp.mpf(2) ** 970:
+ return math.copysign(math.inf, v)
+ if v < 0:
+ return -nearest(-v)
+ lo, hi = down(v), up(v)
+ if lo == hi:
+ return lo
+ dlo, dhi = v - exact(lo), exact(hi) - v
+ if dlo < dhi:
+ return lo
+ if dhi < dlo:
+ return hi
+ # The even one has an even significand, the last bit of its encoding,
+ # subnormal doubles included.
+ return lo if struct.unpack(" 1, x at most the mode, by quadrature.
+
+ Below x the density is at most f(x) exp(-h (x - t)), with h the slope of
+ log f at x (log-concavity), so the pieces [x - 2^k s, x - 2^(k-1) s], with s
+ the smaller of 1/h and the standard deviation, each see the density fall by
+ a bounded factor, and the sum stops once that bound makes the rest
+ negligible.
+ """
+ lb = logbeta(a, b)
+ f = lambda t: mp.exp((a - 1) * mp.log(t) + (b - 1) * mp.log1p(-t) - lb)
+ h = (a - 1) / x - (b - 1) / y
+ sd = mp.sqrt(a * b / ((a + b) ** 2 * (a + b + 1)))
+ s = min(1 / h, sd) if h > 0 else sd
+ points = [x]
+ k = 0
+ while True:
+ p = x - s * mp.mpf(2) ** k
+ if p <= 0:
+ points.append(mp.mpf(0))
+ break
+ points.append(p)
+ if h > 0 and h * (x - p) > 400 + mp.mp.dps * 2.31 and p * mp.exp(-h * (x - p)) < s * mp.mpf(10) ** (-mp.mp.dps - 20):
+ break
+ k += 1
+ points.reverse()
+ return mp.quad(f, points)
+
+
+def series_tails(p, q, z):
+ """(P(Z <= z), P(Z > z)) for Z ~ Beta(p,q), by mpmath's betainc in z. Its
+ hypergeometric series has terms summing in magnitude to at most (1+z)^(q-1)
+ and a value at least (1-z)^(q-1), so the working precision gains that many
+ digits; the complement is formed at that precision, which doubles until the
+ complement has 30 correct digits."""
+ cancel = 0.4343 * 2 * float(mp.atanh(z)) * max(float(q) - 1, 0)
+ extra = int(cancel) + 30
+ while True:
+ with mp.extradps(extra):
+ lower = mp.betainc(p, q, 0, z, regularized=True)
+ upper = 1 - lower
+ if upper > mp.mpf(10) ** (int(cancel) + 30 - mp.mp.dps):
+ return lower, upper
+ extra *= 2
+
+
+# Above this many digits of cancellation the series is not used.
+CANCEL_MAX = 6000
+
+
+def tails(a, b, x, y):
+ """(P(X <= x), P(X > x)) for X ~ Beta(a,b), with y = 1 - x exactly.
+
+ One tail is computed directly and the other as its complement, at a working
+ precision where the complement keeps 30 correct digits; by quadrature, the
+ tail computed directly is the one beyond the mode, so the other is at least
+ about 1/2.
+ """
+ if min(a, b) <= SERIES_MAX:
+ # Evaluate the series on the side with less estimated cancellation.
+ # Either side gives both tails, and the other remains a fallback when
+ # a series argument or its cancellation exceeds the supported range.
+ near = (a, b, x, False) if x <= y else (b, a, y, True)
+ far = (b, a, y, True) if x <= y else (a, b, x, False)
+ choices = []
+ for p, q, z, mirrored in (near, far):
+ if z <= mp.mpf("0.9"):
+ cancel = 0.4343 * 2 * float(mp.atanh(z)) * max(float(q) - 1, 0)
+ if cancel <= CANCEL_MAX:
+ choices.append((cancel, p, q, z, mirrored))
+ if choices:
+ _, p, q, z, mirrored = min(choices, key=lambda choice: choice[0])
+ v, c = series_tails(p, q, z)
+ return (c, v) if mirrored else (v, c)
+ raise RuntimeError(f"no series route for Beta({a},{b}) at {x}")
+ mode = (a - 1) / (a + b - 2)
+ if x <= mode:
+ lower = lower_by_quad(a, b, x, y)
+ return lower, 1 - lower
+ upper = lower_by_quad(b, a, y, x)
+ return 1 - upper, upper
+
+
+def quantile(a, b, u):
+ """The exact u-quantile of Beta(a,b) as (x, y = 1 - x, f(x)), with the
+ smaller of x and y found first, on the log scale, from the smaller of the two
+ probabilities u and 1 - u, each exact."""
+ u = exact(u)
+ half = mp.mpf(1) / 2
+ lower_half, _ = tails(a, b, half, half)
+ if u <= lower_half:
+ p, q, small_is_x, below, above = a, b, True, u, 1 - u
+ else:
+ p, q, small_is_x, below, above = b, a, False, 1 - u, u
+ # z ~ Beta(p,q), z <= 1/2, with P(Z <= z) = below and P(Z > z) = above; t = log z.
+ use_above = above < below
+ def g(t):
+ z = mp.exp(t)
+ lower, upper = tails(p, q, z, 1 - z)
+ if use_above:
+ return mp.log(above) - mp.log(upper)
+ return mp.log(lower) - mp.log(below)
+ if g(mp.log(half)) <= 0:
+ z = half
+ else:
+ # From the mean upward until the tail reaches the target, then downward
+ # from there, so no evaluation lies far beyond the distribution's bulk.
+ mean = p / (p + q)
+ sd = mp.sqrt(p * q / ((p + q) ** 2 * (p + q + 1)))
+ k = 4
+ while g(mp.log(min(half, mean + k * sd))) < 0:
+ k *= 2
+ hi = mp.log(min(half, mean + k * sd))
+ lo = hi - 1
+ while g(lo) >= 0:
+ lo = hi - 2 * (hi - lo)
+ # mpmath's Anderson-Bjorck method keeps the root bracketed in [lo, hi],
+ # and findroot checks the residual at the root it returns.
+ z = mp.exp(mp.findroot(g, (lo, hi), solver="anderson", maxsteps=400))
+ w = 1 - z
+ x, y = (z, w) if small_is_x else (w, z)
+ return x, y, mp.exp(logpdf(a, b, x, y)), small_is_x
+
+
+def certify_bounds(lo, hi, value, a, b, target, use_lower):
+ """Move a non-adjacent endpoint inward if its exact tail misses the tolerance."""
+ adjacent = (down(value), up(value))
+ with mp.workdps(80):
+ p, q, t = exact(a), exact(b), exact(target)
+ delta = mp.mpf(EPS) * t
+ def accepted(v):
+ if v in adjacent:
+ return True
+ x = exact(v)
+ if x == 0:
+ lower, upper = mp.mpf(0), mp.mpf(1)
+ elif x == 1:
+ lower, upper = mp.mpf(1), mp.mpf(0)
+ else:
+ lower, upper = tails(p, q, x, 1 - x)
+ probability = lower if use_lower else upper
+ return abs(probability - t) <= delta
+ while not accepted(lo):
+ lo = math.nextafter(lo, math.inf)
+ while not accepted(hi):
+ hi = math.nextafter(hi, -math.inf)
+ return lo, hi
+
+
+def quantile_row(case):
+ a, b, u = case
+ analytic = a == 1 or b == 1
+ if not (in_domain(a, b) or analytic) or (u < 2.0 ** -53 and not analytic):
+ return [show(a), show(b), show(u), "reject"]
+ with mp.workdps(40):
+ x, y, density, small_is_x = quantile(exact(a), exact(b), u)
+ s = min(exact(u), 1 - exact(u))
+ delta = mp.mpf(EPS) * s
+ x_low, y_high, _, _ = quantile(exact(a), exact(b), exact(u) - delta)
+ x_high, y_low, _, _ = quantile(exact(a), exact(b), exact(u) + delta)
+ xlo, xhi = min(up(x_low), down(x)), max(down(x_high), up(x))
+ ylo, yhi = min(up(y_low), down(y)), max(down(y_high), up(y))
+ # If the entire smaller-share range is below 2^-53, its complement's
+ # probability interval, clipped to [0,1], and the adjacent doubles of
+ # the exact complement fit in [nextafter(1,0), 1].
+ z_high = x_high if small_is_x else y_high
+ near_one_needed = z_high < mp.mpf(2) ** -53
+ if near_one_needed:
+ near_one = (math.nextafter(1.0, 0.0), 1.0)
+ if small_is_x:
+ ylo, yhi = near_one
+ else:
+ xlo, xhi = near_one
+ if exact(u) <= mp.mpf("0.5"):
+ if not (near_one_needed and not small_is_x):
+ xlo, xhi = certify_bounds(xlo, xhi, x, a, b, u, True)
+ if not (near_one_needed and small_is_x):
+ ylo, yhi = certify_bounds(ylo, yhi, y, b, a, u, False)
+ else:
+ target = 1 - exact(u)
+ if not (near_one_needed and not small_is_x):
+ xlo, xhi = certify_bounds(xlo, xhi, x, a, b, target, False)
+ if not (near_one_needed and small_is_x):
+ ylo, yhi = certify_bounds(ylo, yhi, y, b, a, target, True)
+ return [show(a), show(b), show(u), show(xlo), show(xhi), show(ylo), show(yhi),
+ "x" if small_is_x else "y", mp.nstr(x if small_is_x else y, 20),
+ show(nearest(delta / density))]
+
+
+def stick_row(case):
+ """dirichlet as at the constant label u: coordinate i is the remaining
+ stick times the share v_i, the u-quantile of Beta(a_i, a_(i+1) + ...)."""
+ alphas, u = case
+ with mp.workdps(40):
+ rest = mp.mpf(1)
+ rest_low = rest_high = mp.mpf(1)
+ coords = []
+ s = min(exact(u), 1 - exact(u))
+ for i, a in enumerate(alphas[:-1]):
+ later = sum(exact(c) for c in alphas[i + 1:])
+ v, w, _, _ = quantile(exact(a), later, u)
+ delta = mp.mpf(EPS) * s
+ v_low, w_high, _, _ = quantile(exact(a), later, exact(u) - delta)
+ v_high, w_low, _, _ = quantile(exact(a), later, exact(u) + delta)
+ coords.append((rest * v, rest_low * v_low, rest_high * v_high))
+ rest, rest_low, rest_high = rest * w, rest_low * w_low, rest_high * w_high
+ coords.append((rest, rest_low, rest_high))
+ ulps = mp.mpf(2) ** -53
+ cells = []
+ for i, (c, lower, upper) in enumerate(coords):
+ slack = (4 * i + 6) * ulps
+ lo = min(down(lower * (1 - slack)), down(c))
+ hi = max(up(upper * (1 + slack)), up(c))
+ cells += [show(lo), show(hi)]
+ return [",".join(show(a) for a in alphas), show(u)] + cells
+
+
+def law_row(case):
+ """P(X <= x) for X ~ Beta(a,b) at a double x."""
+ a, b, x = case
+ with mp.workdps(40):
+ lower, _ = tails(exact(a), exact(b), exact(x), 1 - exact(x))
+ return [show(a), show(b), show(x), show(nearest(lower))]
+
+
+# Log densities -------------------------------------------------------------
+
+def log_density(case):
+ name, args = case[0], case[1:]
+ with mp.workdps(400):
+ if name == "poissonLogPmf":
+ lam, k = exact(args[0]), args[1]
+ if k == 0:
+ v = -lam
+ else:
+ v = k * mp.log(lam) - lam - mp.loggamma(k + 1)
+ elif name == "gammaLogPdf":
+ k, theta, x = map(exact, args)
+ v = (k - 1) * mp.log(x) - x / theta - k * mp.log(theta) - mp.loggamma(k)
+ elif name == "betaLogPdf":
+ a, b, x = map(exact, args)
+ v = logpdf(a, b, x, 1 - x)
+ elif name == "binomialLogPmf":
+ n, p, k = args
+ p = exact(p)
+ v = (mp.loggamma(n + 1) - mp.loggamma(k + 1) - mp.loggamma(n - k + 1)
+ + (k * mp.log(p) if k > 0 else 0) + ((n - k) * mp.log1p(-p) if n > k else 0))
+ elif name == "dirichletLogPdf":
+ alphas, xs = map(lambda l: [exact(v) for v in l], args)
+ assert sum(xs) == 1, xs
+ v = (mp.loggamma(sum(alphas)) - sum(mp.loggamma(a) for a in alphas)
+ + sum((a - 1) * mp.log(x) for a, x in zip(alphas, xs)))
+ else:
+ raise ValueError(name)
+ return [name, " ".join(render(a) for a in args), show(nearest(v))]
+
+
+def render(arg):
+ if isinstance(arg, list):
+ return "[" + ",".join(show(v) for v in arg) + "]"
+ if isinstance(arg, int):
+ return str(arg)
+ return show(arg)
+
+
+# Cases -----------------------------------------------------------------------
+
+LABELS = [2.0 ** -53, 1e-300, 1e-100, 1e-20, 1e-8, 0.001, 0.25, 0.5, 0.75, 0.999,
+ 1 - 1e-8, 1 - 2.0 ** -30, 1 - 2.0 ** -53]
+PARAMETERS = [1e-10, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 0.0018614780417212304, 0.01, 0.1,
+ 0.5, 1.0, 2.0, 30.0, 53.5, 100.0, 1000.0, 1e4, 1e5, 1e6, 1e7, 1e8, 1e10]
+# Beta(a,1) at the least label of the default generator for a around 53, where
+# the quantile passes 1/2, the mirror cases Beta(1,b), a subnormal quantile, and
+# a large shape.
+BRANCH_CASES = [(a, 1.0, 2.0 ** -53) for a in [52.0, 53.0, 54.0, 100.0, 1000.0, 1e4]] + [
+ (1.0, b, 1 - 2.0 ** -53) for b in [52.0, 53.0, 54.0, 100.0, 1000.0, 1e4]] + [
+ (0.0018614780417212304, 1.0, 0.25), (1e16, 1e16, 0.25), (1e16, 1e16, 0.75)]
+
+
+def in_domain(a, b):
+ """The documented domain of dirichlet [a,b] and the general case of beta a b
+ (src/LazyPPL/Distributions.hs); beta also takes Beta(a,1) and Beta(1,b)."""
+ return min(a, b) >= PARAM_LEAST and a + b <= SUM_MAX
+
+
+def edge_cases():
+ """The domain's corners and the doubles just outside them, at every label:
+ the least parameter beside a sum at the bound, two halves of the bound, and
+ pairs one double below the least parameter or one double above the sum."""
+ least, most = PARAM_LEAST, SUM_MAX
+ inside = [(least, least), (least, 1.0), (least, most - least), (1.0, most - 1.0), (most / 2, most / 2)]
+ below = math.nextafter(least, 0)
+ outside = [(below, 1.0), (below, most - least), (most / 2, math.nextafter(most / 2, math.inf))]
+ pairs = inside + outside
+ return [(a, b, u) for a, b in pairs + [(b, a) for a, b in pairs if a != b] for u in LABELS]
+
+
+def random_cases(count, seed=20261101):
+ """Parameters drawn log-uniformly inside the domain, each with a label of LABELS, so the
+ check covers more than the grid's points."""
+ import random
+ rng = random.Random(seed)
+ cases = []
+ while len(cases) < count:
+ a = 10 ** rng.uniform(math.log10(PARAM_LEAST), math.log10(SUM_MAX))
+ b = 10 ** rng.uniform(math.log10(PARAM_LEAST), math.log10(SUM_MAX))
+ if in_domain(a, b):
+ cases.append((float(f"{a:.6g}"), float(f"{b:.6g}"), rng.choice(LABELS)))
+ return cases
+
+
+def quantile_cases():
+ grid = [(a, b, u) for a in PARAMETERS for b in PARAMETERS for u in LABELS]
+ extra = BRANCH_CASES + edge_cases() + random_cases(400)
+ return list(dict.fromkeys(grid + extra))
+
+
+def stick_cases(least, sum_max):
+ third = sum_max / 3
+ vectors = [[1.0, 2.0, 3.0], [3.0, 2.0, 1.0], [2.0, 3.0, 1.0],
+ [0.5, 0.5, 0.5, 0.5], [0.001, 1.0, 5.0], [5.0, 1.0, 0.001], [1.0, 5.0, 0.001],
+ [third, third, third], [least, least, least], [least, sum_max - 2 * least, least],
+ [30.0, 0.01, 0.5, 2.0]]
+ return [(v, u) for v in vectors for u in [2.0 ** -53, 0.25, 0.5, 0.75, 1 - 2.0 ** -53]]
+
+
+def law_cases(shapes):
+ """Points for the law checks: 1201 points from the mean minus six standard
+ deviations to the mean plus six where both parameters exceed SERIES_MAX
+ (the law is then close to normal), and the quantiles at the levels k/1000
+ otherwise. Closed forms select the points for the stick laws; law_row still
+ computes their expected distribution functions independently with betainc.
+ """
+ rows = []
+ for a, b in shapes:
+ with mp.workdps(40):
+ a_, b_ = exact(a), exact(b)
+ if min(a, b) > SERIES_MAX:
+ mean = a_ / (a_ + b_)
+ sd = mp.sqrt(a_ * b_ / ((a_ + b_) ** 2 * (a_ + b_ + 1)))
+ xs = {nearest(mean + sd * mp.mpf(z) / 100) for z in range(-600, 601)}
+ elif b == 1:
+ xs = {nearest(mp.exp(mp.log(exact(k / 1000)) / a_)) for k in range(1, 1000)}
+ elif a == 1:
+ xs = {nearest(-mp.expm1(mp.log1p(-exact(k / 1000)) / b_)) for k in range(1, 1000)}
+ else:
+ xs = {nearest(quantile(a_, b_, k / 1000)[0]) for k in range(1, 1000)}
+ rows += [(a, b, x) for x in sorted(xs) if 0 < x < 1]
+ return rows
+
+
+def density_cases():
+ cases = []
+ lams = [1e-320, 1e-300, 1e-10, 0.5, 1.0, 3.0, 10.0, 1e6, 1e16, 1e32, 1e100, 1e300, 3e307, 1e308, MAXD]
+ ks = [0, 1, 2, 10, 1000000, 1000001, 10 ** 16, 2 ** 53 + 1, 10 ** 32, 10 ** 100, 10 ** 300, 3 * 10 ** 307, 10 ** 308]
+ for lam in lams:
+ cases += [("poissonLogPmf", lam, k) for k in ks]
+ t = int(lam)
+ if t > 1:
+ spread = max(1, math.isqrt(t))
+ cases += [("poissonLogPmf", lam, k) for k in [t - 3 * spread, t - 1, t, t + 1, t + 3 * spread]]
+ cases += [("poissonLogPmf", 1.25e307, int(1e308)), ("poissonLogPmf", 1e308, 2 * 10**308),
+ ("poissonLogPmf", MAXD, int(MAXD) + 2**971), ("poissonLogPmf", MAXD, 3 * 10**308)]
+ shapes = [1e-300, 1e-10, 0.5, 1.0, 2.0, 1000.0, 1e16, 1e300, 1e308]
+ scales = [1e-300, 1e-10, 1.0, 8.0, 1e10, 1e300]
+ points = [1e-320, 1e-300, 1e-10, 1.0, 3.0, 1e10, 1e300, 1e308]
+ cases += [("gammaLogPdf", k, th, x) for k in shapes for th in scales for x in points]
+ cases += [("gammaLogPdf", 3e307, 1.0, 3e307), ("gammaLogPdf", 1e308, 8.0, 1e308), ("gammaLogPdf", 1e16, 1.0, 1e16)]
+ # Near the centre with large parameters, where a rounded x / theta, n * p or
+ # (a + b - 2) * x moves the deviance: z standard deviations from the centre.
+ zs = [0.5, 3.0, 30.0]
+ for k in [1e6, 1e8, 1e12, 1e20, 1e100, 1e300]:
+ for theta in [3.0, 1e-7]:
+ cases += [("gammaLogPdf", k, theta, theta * (k - 1 + z * math.sqrt(k))) for z in zs]
+ cases += [("gammaLogPdf", 1e300, 1e-300, 1.0), ("gammaLogPdf", 1e308, 0.5, 1e308),
+ ("gammaLogPdf", MAXD, 0.9, MAXD)]
+ for n in [10 ** 6, 10 ** 8, 10 ** 12, 2 ** 62]:
+ for p in [0.3, 1 / 3, 0.7]:
+ sd = math.sqrt(n * p * (1 - p))
+ cases += [("binomialLogPmf", n, p, round(n * p + z * sd)) for z in zs]
+ for a, b in [(1e6, 3e6), (1e8, 3e8), (1e12, 1e12 + 5), (1e100, 2e100), (1e300, 3e300)]:
+ mode = (a - 1) / (a + b - 2)
+ sd = math.sqrt(a) * math.sqrt(b) / (a + b) / math.sqrt(a + b + 1)
+ cases += [("betaLogPdf", a, b, mode + z * sd) for z in zs]
+ concentrations = [1e-300, 1e-100, 1e-10, 1e-3, 0.5, 1.0, 1.5, 2.0, 3.0, 10.0, 1000.0, 1e8, 1e16, 1e100, 1e300, 8e307]
+ xs = [1e-300, 1e-20, 2.0 ** -53, 1e-3, 0.25, 0.5, 0.75, 1 - 2.0 ** -53]
+ cases += [("betaLogPdf", a, b, x) for a in concentrations for b in concentrations for x in xs]
+ cases += [("betaLogPdf", 1.0, 1e20, 1e-20), ("betaLogPdf", 2.0, 2.0, 0.5)]
+ ns = [1, 10, 1000, 1000000, 10 ** 16, 2 ** 62, 2 ** 63 - 1]
+ ps = [0.0, 1e-300, 1e-16, 1e-3, 0.3, 0.5, 0.7, 1 - 2.0 ** -53, 1.0]
+ for n in ns:
+ for p in ps:
+ for k in sorted({0, 1, n // 2, n - 1, n, min(n, max(0, round(n * p)))}):
+ if (p == 0 and k > 0) or (p == 1 and k < n):
+ continue
+ cases.append(("binomialLogPmf", n, p, k))
+ cases += [("dirichletLogPdf", al, x) for al, x in [
+ ([1e200, 1e200], [0.5, 0.5]), ([1e16, 1e16], [0.5, 0.5]), ([1e300, 1e300], [0.5, 0.5]),
+ ([1.0, 1.0, 1.0], [0.25, 0.25, 0.5]), ([2.0, 3.0, 5.0], [0.125, 0.375, 0.5]),
+ ([0.001, 0.001, 0.002], [0.25, 0.25, 0.5]), ([1e-300, 1.0, 1e300], [0.25, 0.25, 0.5]),
+ ([0.5, 0.5, 0.5, 0.5], [0.0625, 0.1875, 0.25, 0.5]), ([1e8, 3e8, 4e8], [0.125, 0.375, 0.5]),
+ ([1e8, 3e8, 4e8], [0.125 + 2.0 ** -20, 0.375 - 2.0 ** -20, 0.5]),
+ ([1e12, 3e12, 4e12], [0.125 + 2.0 ** -24, 0.375 - 2.0 ** -24, 0.5])]]
+ # Each case once, at its first place; repr tells the cases apart, lists included.
+ unique = {}
+ for case in cases:
+ unique.setdefault(repr(case), case)
+ return list(unique.values())
+
+
+# Output ----------------------------------------------------------------------
+
+def write(name, header, rows):
+ path = os.path.join(HERE, name)
+ with open(path, "w") as out:
+ out.write(f"# Written by test/reference/reference.py with mpmath {MPMATH_VERSION}.\n")
+ out.write("# " + "\t".join(header) + "\n")
+ for row in rows:
+ out.write("\t".join(row) + "\n")
+ print(f"wrote {path}: {len(rows)} rows", flush=True)
+
+
+def self_checks():
+ """The tail and quantile routines against closed forms and each other."""
+ with mp.workdps(40):
+ def close(v, ref, name, rel=mp.mpf("1e-25")):
+ if abs(v - ref) > rel * abs(ref):
+ sys.exit(f"self-check failed: {name}: {v} against {ref}")
+ for a, x in [(0.001, 1e-300), (30.0, 0.7), (1e8, 0.9999999), (2.5, 0.5)]:
+ a_, x_ = exact(a), exact(x)
+ lower, upper = tails(a_, mp.mpf(1), x_, 1 - x_)
+ close(lower, x_ ** a_, f"Beta({a},1) lower tail at {x}")
+ close(upper, -mp.expm1(a_ * mp.log(x_)), f"Beta({a},1) upper tail at {x}")
+ lower, upper = tails(mp.mpf(1), a_, x_, 1 - x_)
+ close(upper, (1 - x_) ** a_, f"Beta(1,{a}) upper tail at {x}")
+ a, b, x = mp.mpf("1e-8"), mp.mpf(10000), mp.mpf("0.5")
+ _, upper = tails(a, b, x, 1 - x)
+ direct_upper = mp.betainc(b, a, 0, 1 - x, regularized=True)
+ close(upper, direct_upper, "asymmetric Beta upper tail")
+ lower, _ = tails(mp.mpf("0.5"), mp.mpf("0.5"), mp.mpf("0.3"), mp.mpf("0.7"))
+ close(lower, 2 / mp.pi * mp.asin(mp.sqrt(mp.mpf("0.3"))), "Beta(1/2,1/2) at 0.3")
+ # The two methods agree where both apply.
+ # The two methods agree where both apply, including betainc's far side
+ # (an argument near 1) for a large parameter.
+ for a, b, x in [(150.0, 150.0, 0.45), (150.0, 400.0, 0.2), (120.0, 130.0, 0.3),
+ (101.5, 1e6, 7e-5), (101.5, 1e6, 1.4e-4)]:
+ a_, b_, x_ = exact(a), exact(b), exact(x)
+ mode = (a_ - 1) / (a_ + b_ - 2)
+ by_quad = lower_by_quad(a_, b_, x_, 1 - x_) if x_ <= mode else 1 - lower_by_quad(b_, a_, 1 - x_, x_)
+ by_series = series_tails(a_, b_, x_)[0]
+ by_series_far = series_tails(b_, a_, 1 - x_)[1]
+ close(by_quad, by_series, f"quad against betainc for Beta({a},{b}) at {x}")
+ close(by_quad, by_series_far, f"quad against betainc's far side for Beta({a},{b}) at {x}")
+ # Closed forms: Beta(a,1) has the quantile u^(1/a), Beta(1,b) has 1 - (1-u)^(1/b).
+ for a, u in [(1000.0, 1e-300), (100.0, 2.0 ** -53), (0.001, 1e-300), (1e8, 1 - 2.0 ** -53), (30.0, 0.25)]:
+ x, y, _, _ = quantile(exact(a), mp.mpf(1), u)
+ close(x, exact(u) ** (1 / exact(a)), f"Beta({a},1) quantile at {u}", mp.mpf("1e-30"))
+ x, y, _, _ = quantile(mp.mpf(1), exact(a), u)
+ close(y, (1 - exact(u)) ** (1 / exact(a)), f"Beta(1,{a}) complement of the quantile at {u}", mp.mpf("1e-30"))
+ for a, b, u in [(0.5, 3.0, 1e-300), (1e6, 1e6, 0.25), (1.0, 0.001, 0.25), (200.0, 1.0, 2.0 ** -53)]:
+ x, y, _, small_is_x = quantile(exact(a), exact(b), u)
+ lower, upper = tails(exact(a), exact(b), x, y)
+ target = exact(u) if small_is_x else 1 - exact(u)
+ close(lower if small_is_x else upper, target, f"quantile of Beta({a},{b}) at {u}", mp.mpf("1e-30"))
+ row = quantile_row((1000.0, 1.0, 2.0 ** -53))
+ lo, hi = map(float, row[3:5])
+ bad = 0.9639298077416072
+ u = mp.mpf(2) ** -53
+ assert abs(exact(bad) ** 1000 - u) > mp.mpf(EPS) * u
+ assert not lo <= bad <= hi, "out-of-tolerance endpoint accepted"
+ quantile_value = u ** (mp.mpf(1) / 1000)
+ assert all(lo <= v <= hi for v in (down(quantile_value), up(quantile_value)))
+ assert lo <= math.exp(math.log(float(u)) / 1000) <= hi
+ sparse = quantile_row((0.001, 0.001, 0.999))
+ assert list(map(float, sparse[3:5])) == [math.nextafter(1.0, 0.0), 1.0]
+ print("self-checks passed", flush=True)
+
+
+def main():
+ parser = argparse.ArgumentParser()
+ parser.add_argument("--jobs", type=int, default=os.cpu_count())
+ parser.add_argument("--only", choices=["quantiles", "sticks", "laws", "densities"], action="append")
+ parser.add_argument("--probe", action="store_true", help="time a few quantile rows and stop")
+ args = parser.parse_args()
+ only = set(args.only or ["quantiles", "sticks", "laws", "densities"])
+ self_checks()
+ if args.probe:
+ import time
+ for case in [(0.5, 3.0, 0.25), (1e-10, 1e-10, 0.25), (100.0, 1.0, 2.0 ** -53), (1e8, 53.5, 0.25),
+ (53.5, 1e8, 1e-300), (1e4, 1e4, 0.25), (1e6, 1e6, 1e-300), (1e8, 1e8, 0.75),
+ (1e10, 1e10, 0.25), (1e16, 1e16, 0.25)]:
+ start = time.time()
+ row = quantile_row(case)
+ print(f"{time.time() - start:7.2f}s", "\t".join(row), flush=True)
+ return
+ with Pool(args.jobs) as pool:
+ if "densities" in only:
+ write("log-densities.tsv", ["function", "arguments", "exact log density, nearest double"],
+ pool.map(log_density, density_cases(), chunksize=4))
+ if "quantiles" in only:
+ write("beta-quantiles.tsv",
+ ["a", "b", "u", "x low or reject", "x high", "1-x low", "1-x high", "smaller side",
+ "smaller side exact", "linearised half-width (informational)"],
+ pool.map(quantile_row, quantile_cases(), chunksize=1))
+ if "sticks" in only:
+ write("dirichlet-sticks.tsv", ["alphas", "u", "coordinate 1 low", "coordinate 1 high", "..."],
+ pool.map(stick_row, stick_cases(PARAM_LEAST, SUM_MAX), chunksize=1))
+ if "laws" in only:
+ write("beta-laws.tsv", ["a", "b", "x", "P(X <= x), nearest double"],
+ pool.map(law_row, law_cases(LAW_SHAPES), chunksize=8))
+
+
+# The supported domain of the Beta and Dirichlet samplers, measured with the
+# table of this script (src/LazyPPL/Distributions.hs, betaLeast). The edge
+# cases, the random cases, two of the stick vectors and the law shape
+# (1000,1000), the largest symmetric one inside, come from it.
+PARAM_LEAST = 1e-3
+SUM_MAX = 2000.0
+# Beta(1,b) and Beta(a,1) at a concentration of 1e-4 and of 1e4: the sticks of
+# the Dirichlet process (Beta(1,alpha)) and of newRestaurantS (Beta(alpha,1)).
+LAW_SHAPES = [(2.5, 3.5), (1000.0, 1000.0), (1.0, 1e-4), (1.0, 1e4), (1e-4, 1.0), (1e4, 1.0)]
+
+if __name__ == "__main__":
+ main()