The Multivariate Posterior Distribution

Abstract

Irénée Jules Bienaymé (1796–1878) proposes to generalize Laplace’s inverse probability analysis of the binomial. Using the principle of inverse probability on the multinomial he gets the posterior distribution

$$ p_n \left( {\theta _1 ,...,\theta _k |n_1 ,...,n_k } \right)\alpha \theta _1^{n_1 } ...\theta _k^{n_k } ,0 < \theta _i < 1,\sum \theta _i = 1, $$

(1)

where the ns are nonnegative integers and Σn _i = n. In normed form this distribution is today called the Dirichlet distribution. The posterior mode is h _i = n _i /n, Σh _i = 1.