A technical note on the Dirichlet-Multinomial model: The Dirichlet Distribution as the canonically constructed conjugate prior

This short note contains an explicit proof of the Dirichlet distribution being the conjugate prior to the Multinomial sample distribution as resulting from the general construction method described, e.g., in Bernardo and Smith (2000). The well-known Dirichlet-Multinomial model is thus shown to fit into the framework of canonical conjugate analysis (Bernardo and Smith 2000, Prop. 5.6, p. 273), where the update step for the prior parameters to their posterior counterparts has an especially simple structure. This structure is used, e.g., in the Imprecise Dirichlet Model (IDM) by Walley (1996), a simple yet powerful model for imprecise Bayesian inference using sets of Dirichlet priors to model vague prior knowledge, and furthermore in other imprecise probability models for inference in exponential families where sets of priors are considered.