Abstract

We provide a new characterization of the Dirichlet distribution. Let ` ij , 1 i k; 1 j n, be positive random variables that sum to unity. Define ` i\Delta = P n j=1 ` ij , `I \Delta = f` i\Delta g k\Gamma1 i=1 , ` jji = ` ij = P j ` ij , and ` Jji = f` jji g n\Gamma1 j=1 . We prove that if f`I \Delta ; ` Jj1 ; : : : ; ` Jjk g are mutually independent and f` \DeltaJ ; ` Ij1 ; : : : ; ` Ijn g are mutually independent (where ` \DeltaJ and ` Ijj are defined analogously), and assuming strictly positive pdfs, then the pdf of ` ij is Dirichlet. 1 Introduction Suppose s and t are two discrete random variables having finite domains, fs i g k i=1 and ft j g n j=1 , respectively. We wish to infer the joint probability p(s; t) from a sample of pairs of values (s i ; t j ) of s and t. The standard Bayesian approach to this statistical inference problem is to associate with p(s i ; t j ) a parameter ` ij (often called the multinomial parameter), assign f` ij j1 i k; 1 j ng a p...

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