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Probability Distribution Monad

Makes it easy to create, manipulate and sample probability distributions.

For example, here's how you would code up the following problem: You are given either a fair coin or a biased coin with equal probability. If you flip it 5 times and it comes up heads each time, what is the probability you have the fair coin?

case class Trial(haveFairCoin: Boolean, flips: List[Coin])
def bayesianCoin(nflips: Int): Distribution[(Int, List[Coin])] = {
  for {
    haveFairCoin <- tf()
    c = if (haveFairCoin) coin else biasedCoin(0.9)
    flips <- c.repeat(nflips)
  } yield Trial(haveFairCoin, flips)

bayesianCoin(5).given(_.flips.forall(_ == H)).pr(_.haveFairCoin)

Or: You repeatedly roll a 6-sided and keep a running sum. What is the probability the sum reaches exactly 30?

def dieSum(rolls: Int): Distribution[List[Int]] = {
  markov(rolls, List(0))(runningSum => for {
    d <- die
  } yield (d + runningSum.head) :: runningSum)

dieSum(30).pr(_ contains 30)

Distribution.scala contains code for creating and manipulating probability distributions. Built-in distributions include:

  • uniform continuous
  • uniform discrete (including die and fair coin)
  • weighted discrete (biased coin, uses the alias method)
  • normal
  • poisson
  • cauchy
  • chi2

Methods for manipulating distributions include:

  • adding (convolution), subracting (cross-correlation), multiplying and dividing distributions
  • producing joint distributions from single distributions
  • conditional distributions (amounts to a filter)
  • mapping and flatMapping values inside the distribution
  • creating Markov chains
  • finding the probability of arbitrary predicates, conditional probabililty
  • finding expected values
  • sampling, histogram

Examples.scala contains some example uses, and possibly a RISK simulator.