A characterization of the Dirichlet distribution through global and local parameter independence

We provide a new characterization of the Dirichlet distribution. Let θᵢⱼ, 1 ≤ i ≤ k, 1 ≤ j ≤ n, be positive random variables that sum to unity. Define θi · = Σⱼ₌₁ⁿ θᵢⱼ, θI · = θi ·ᵢ₌₁ᵏ⁻¹, θj|i = θᵢⱼ/ Σⱼ θᵢⱼ and θJ|i = {θj|i}ⱼ₌₁ⁿ⁻¹$. We prove that if ${θI ·, θJ|1, , θJ|k}$ are mutually independent and ${θ· J, θI|1, , θI|n}$ are mutually independent (where $θ· J$ and $θI|j$ are defined analogously, and each parameter set has a strictly positive pdf, then the pdf of $θᵢⱼ$ 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.