Bayesian networks on dirichlet distributed vectors

Exact Bayesian network inference exists for Gaussian and multinomial distributions. For other kinds of distributions, approximations or restrictions on the kind of inference done are needed. In this paper we present generalized networks of Dirichlet distributions, and show how, using the two-parameter Poisson-Dirichlet distribution and Gibbs sampling, one can do approximate inference over them. This involves integrating out the probability vectors but leaving auxiliary discrete count vectors in their place. We illustrate the technique by extending standard topic models to "structured" documents, where the document structure is given by a Bayesian network of Dirichlets.