LazyPPL
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LazyPPL

Laziness for Bayesian nonparametrics.

LazyPPL is a Haskell library for Bayesian probabilistic programming.

It supports lazy use of probability, and we provide new Metropolis-Hastings algorithms to allow this.

LazyPPL is inspired by recent ideas in synthetic probability theory and synthetic measure theory, such as quasi-Borel spaces and Markov categories. Laziness appears to be a good paradigm for non-parametric statistics.

LazyPPL is inspired by many other languages, including Church, Anglican, and MonadBayes. Several aspects are now incorporated into MonadBayes (see here).

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01∞

A Poisson process on , sampled lazily. Each draw becomes a gap .

Haskell source

Haskell
splice :: Prob [Double] -> Prob (Double -> Double) -> Prob (Double -> Double)splice pointProcess randomFun =  do    xs <- pointProcess    fs <- mapM (const randomFun) xs    default_f <- randomFun    let h :: [(Double, Double -> Double)] -> Double -> Double        h [] x = default_f x        h ((a, f) : xfs) x | x <= a = f x        h ((a, f) : xfs) x | x > a = h xfs x    return (h (zip xs fs))poissonPP :: Double -> Double -> Prob [Double]poissonPP lower rate =  do    step <- exponential rate    let x = lower + step    xs <- poissonPP x rate    return (x : xs)regress 0.1 (splice (poissonPP 0 0.1) linear)

Piecewise linear regression

Posterior functions
Ready.
Piecewise posterior functions and observationsRun regression to draw piecewise linear posterior functions.
  • log-likelihood low high
  • Observations
Numerical results
Linear & Piecewise Regression
Gaussian Process Regression
Wiener Process Regression
Non-parametric Clustering
Infinite Relational Model
Program Induction
Feature Finding Using an Indian Buffet Process
Graph Inference

Worked examples

All examples

Laziness lets us specify non-parametric models cleanly: processes or infinite-dimensional functions only have their finite explored portion materialised.

Posterior piecewise regression functions and observations.Posterior piecewise regression functions and observations.

Linear & Piecewise Regression

Start simple with Bayesian linear regression, then extend to piecewise models using a Poisson point process.

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Posterior Wiener-process functions and observations.Posterior Wiener-process functions and observations.

Wiener Process Regression

Model time series with Wiener process priors for continuous, nowhere-differentiable random functions.

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Three inferred program curves and the observations they explain.Three inferred program curves and the observations they explain.
Programs
  • 1
  • 2
  • 3

Program Induction

Infer arithmetic expressions that explain observed data using probabilistic program induction.

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Observations assigned to numbered clusters over their inferred densities.Observations assigned to numbered clusters over their inferred densities.
Clusters
  • Cluster 1
  • Cluster 2

Non-parametric Clustering

Dirichlet Process mixture models that discover the number of clusters automatically.

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Two-Monad Architecture

LazyPPL provides two monads:

  • Prob a

    probability measures, supporting probability measures such as uniform :: Prob Double, normal :: Double -> Double -> Prob Double, bernoulli :: Double -> Prob Bool. This is lazy, in other words it is an affine monad.

  • Meas a

    unnormalized measures, as used in Monte Carlo simulations for Bayesian statistics. There are two key functions:

    • sample :: Prob a -> Meas a, which samples from a probability measure;
    • score :: Double -> Meas (), which weights a measure by a given value, typically coming from a likelihood function.
API reference Papers Simple example