A Characterization of the Dirichlet Distribution Through Global and Local Independence

Local computations with probabilities on graphical structures and their application to expert systems. A natural question to ask is whether global independence alone implies a joint Dirichlet pdf for f ij g. This question is particularly interesting in light of the analysis of decomposable graphical models given by Dawid and Lauritzen, 1993]. Dawid and Lauritzen term a pdf that satisses global independence a strong hyper-Markov law and show the importance of such laws in the analysis of decomposable graphical models. We now show that the class of strong hyper-Markov laws is larger than the Dirichlet class. When n = k = 2, and using the notations of Section 4, the new functional equation can be written as follows: f 0 (y)g(z; w) = g 0 (x)f(yz x ; y(1 ? z) 1 ? x) (94) where x = yz + (1 ? y)w. Note that Eq. 11 is obtained from this equation by setting g(z; w) = g 1 (z)g 2 (w) and f(t 1 ; t 2) = f 1 (t 1)f 2 (t 2). These equalities correspond to local independence which is assumed in earlier sections. Let f U be a joint pdf of f ij g given by f U (f ij g) = K 2 4 2 Y i=1 2 Y j=1 ij?1 ij 3 5 H(11 22 12 21) (95) where K is the normalization constant, ij are positive constants and H is an arbitrary positive integrable function. That this pdf satisses global independence can be easily veriied. It can in fact be shown, by solving Eq. 94, that every positive strong Hyper Markov law can be written in this form (when n = 2 and k = 2). This solution includes the Dirichlet family as a subclass. A full characterization of strong hyper Markov laws will be given in a follow-up article. Acknowledgment We thank, J. Acz el, M. Israeli, and M. Ungarish for valuable comments. We thank S. Altschuler and L. Wu for their help with the proof of Lemma 3. Suppose x 1 ; : : :; x m are m discrete random variables having nite domains. With each of the n j possible assignments of values to x j , j = 1; : : :; m, we associate a multinomial parameter i1;:::;im. Analogously to the case m = 2 discussed in previous sections, let g are mutually independent and f …