An Expectation Formula for the Multivariate Dirichlet Distribution

Suppose that the random vector (X1, …, Xq) follows a Dirichlet distribution on Rq+ with parameter (p1, …, pq)∈Rq+. For f1, …, fq>0, it is well-known that E(f1X1+…+fqXq)−(p1+…+pq)=f−p11…f−pqq. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Ωr denote the cone of all r×r positive-definite real symmetric matrices. For x∈Ωr and 1⩽j⩽r, let detj x denote the jth principal minor of x. For s=(s1, …, sr)∈Rr, the generalized power function of x∈Ωr is the function Δs(x)=(det1 x)s1−s2 (det2 x)s2−s3…(detr−1 x)sr−1−sr (detr x)sr; further, for any t∈R, we denote by s+t the vector (s1+t, …, sr+t). Suppose X1, …, Xq∈Ωr are random matrices such that (X1, …, Xq) follows a multivariate Dirichlet distribution with parameters p1, …, pq. Then we evaluate the expectation E[Δs1(X1)…Δsq(Xq) Δs1+…+sq+p((a+f1X1+…+fqXq)−1)], where a∈Ωr, p=p1+…+pq, f1, …, fq>0, and s1, …, sq each belong to an appropriate subset of Rr+. The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xjʹs are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones.