A characterization of the Dirichlet distribution through global and local parameter independence
We provide a new characterization of the Dirichlet distribution. Let θ ij , 1 ≤ i < k, 1 ≤ j ≤ n, be positive random variables that sum to unity. Define θ i = Σ j n = 1 θ ij , θ 1 = {θ i } i=1 k=1 , θ j/i = θ ij /Σj θ ij and θ j/i = {θ j/i } j=1 n-1 . We prove that if {θi, θ j/1 ,..., θ j/k } are mutually independent and {θ. j , θ l/1 ,..., θ I/n } are mutually independent (where θ. j and θ I/j are defined analogously), and each parameter set has a strictly positive pdf, then the pdf of θ ij is Dirichlet. This characterization implies that under assumptions made by several previous authors for selecting a Bayesian network structure out of a set of candidate structures, a Dirichlet prior on the parameters is inevitable.