Deriving a probability density calculator (functional pearl)

Abstract

Given an expression that denotes a probability distribution, often we want a corresponding density function, to use in probabilistic inference. Fortunately, the task of finding a density has been automated. It turns out that we can derive a compositional procedure for finding a density, by equational reasoning about integrals, starting with the mathematical specification of what a density is. Moreover, the density found can be run as an estimation algorithm, as well as simplified as an exact formula to improve the estimate.