A Technical Note on the Dirichlet-Multinomial Model

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.