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

Abstract

We provide a new characterization of the Dirichlet distribution. Let $\theta_{ij}, 1 \leq i \leq k, 1 \leq j \leq n$, be positive random variables that sum to unity. Define $\theta_{i \cdot} = \Sigma_{j=1}^n \theta_{ij}, \theta_{I \cdot} = {\theta_{i \cdot}_{i=1}^{k-1}, \theta_{j|i} = \theta_{ij}/ \Sigma_j \theta_{ij}$ and \theta_{J|i} = {\theta_{j|i}}_{j=1}^{n-1}$. We prove that if ${\theta_{I \cdot}, \theta_{J|1}, \dots, \theta_{J|k}}$ are mutually independent and ${\theta_{\cdot J}, \theta_{I|1}, \dots, \theta_{I|n}}$ are mutually independent (where $\theta_{\cdot J}$ and $\theta_{I|j}$ are defined analogously, and each parameter set has a strictly positive pdf, then the pdf of $\theta_{ij}$ 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.