Exact Bayesian inference on mixtures

Following the ICMS workshop on mixtures that took place in Edinburgh last March, Kerrie Mengersen, Mike Titterington and myself will soon publish as editors a book made of 15 contributions from the invited speakers. In addition, Mike Titterington wrote a chapter on “The EM Algorithm, Variational Approximations and Expectation Propagation for Mixtures”, while Kerrie Mengersen and I wrote a chapter on “Exact Bayesian Analysis of Mixtures” that we also posted on arXiv. This chapter was originally to take place in a book we had planned on mixtures, a book that we never found the time to complete. The main point in this chapter is that there are specific mixtures, namely Poisson and multinomial, for which the complexity remains manageable to conduct exact computations for a while. This point is quite related with an earlier 2005 paper of Paul Fearnhead in Statistics and Computing.