deanie is an embedded probabilistic programming language. It can be used to denote, sample from, and perform inference on probabilistic programs.

Programs are written in a straightforward monadic style:

mixture :: Double -> Double -> Program Double
mixture a b = do
  p      <- beta a b
  accept <- bernoulli p
  if   accept
  then gaussian (negate 2) 0.5
  else gaussian 2 0.5

You can sample from them by first converting them into an RVar from random-fu:

> sample (rvar (mixture 1 3))

Sample many times from models using standard monadic combinators like 'replicateM':

> replicateM 1000 (sample (rvar (mixture 1 3)))

Or convert them to measures using a built-in interpreter:

> let nu = measure (mixture 1 3)
> let f = cdf nu

You can perform inference on models using rejection or importance sampling, or use a simple, stateful Metropolis backend. Here's a simple beta-bernoulli model, plus some observations to condition on:

betaBernoulli :: Double -> Double -> Program Bool
betaBernoulli a b = do
  p <- beta a b
  bernoulli p

observations :: [Bool]
observations = [True, True, False, True, False, False, True, True, True]

Here's one way to encode a posterior via rejection sampling:

rposterior :: Double -> Double -> Program Double
rposterior a b =
    grejection
      (\xs ys -> count xs == count ys)
      observations (beta a b) bernoulli
  where
    count = length . filter id

Here's another, via importance sampling:

iposterior :: Double -> Double -> Program (Double, Double)
iposterior a b =
  importance observations (beta a b) logDensityBernoulli

There are also some Monte Carlo convenience functions provided, such as a weighted average for weighted samples returned via importance sampling:

> samples <- replicateM 1000 (sample (rvar (iposterior 1 1)))
> print (mcw samples)
0.6369246537796793

You can read about some of the theory and ideas behind this kind of language in some blog posts I've written.

• Encoding Statistical Independence, Statically
• A Simple Embedded Probabilistic Programming Language
• Comonadic MCMC
• Foundations of the Giry Monad
• Implementing the Giry Monad
• The Applicative Structure of the Giry Monad