Beschreibung:
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·= ∑n j=1θij, θI·= {θi·}k-1 i=1, θj∣ i= θij/∑jθijand θJ∣ i= {θj∣ i}n-1 j=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· Jand θI∣ jare defined analogously), and each parameter set has a strictly positive pdf, then the pdf of θijis 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.