Dirichlet

Dirichlet( alpha,I )

A Dirichlet distribution with parameters alpha_i>0.

Each sample of a Dirichlet distribution produces a random vector whose elements sum to 1. It is commonly used to represent second order probability information.

The Dirichlet distribution has a density given by

k * Product( X^(alpha-1), I)

where k is a normalization factor equal to

GammaFn( Sum(alpha,I )) / Sum(GammaFn(alpha),I)

The parameters, alpha, can be interpreted as observation counts. The mean is given by the relative values of alpha (normalized to 1), but the variance narrows as the alphas get larger, just as your confidence in a distribution would narrow as you get more samples.

The Dirichlet lends itself to easy Bayesian updating. If you have a prior of alpha0, and you observe N

Library

Multivariate Distributions.ana