2.9 ( ) www . In this exercise, we prove the normalization of the Dirichlet distribution

(2.38) using induction. We have already shown in Exercise 2.5 that the beta distribution, which is a special case of the Dirichlet for M = 2, is normalized.

We now assume that the Dirichlet distribution is normalized for M − 1 variables and prove that it is normalized for M variables. To do this, consider the Dirichlet distribution over M variables, and take account of the constraint

M k=1 μk = 1 by eliminating μM, so that the Dirichlet is written pM(μ1, . . . , μM−1) = CM M−1 k=1

μαk−1 k



1 −

M−1 j=1

μj

αM−1

(2.272)

and our goal is to find an expression for CM. To do this, integrate over μM−1, taking care over the limits of integration, and then make a change of variable so that this integral has limits 0 and 1. By assuming the correct result for CM−1 and making use of (2.265), derive the expression for CM.