Bayesian Nonparametrics and Mixture Modelling

This introductory chapter is aimed at post-graduate students, not necessarily with a strong mathematical background, but with knowledge of the fundamentals of probability and statistics. It is based on the author’s own research and other sources referenced within. We start with an introduction of Bayesian nonparametrics and the Dirichlet process. Parts of this introduction are based on lecture notes by Professor Tony O’Hagan (Lecture notes on Bayesian inference. University of Nottingham, 1996). We continue with an overview of Bayesian mixture modelling, considering mixture models with a finite number of components, where this number can be fixed or random. We then proceed to discuss the Dirichlet process mixture model where an infinite number of components is assumed. Relevant MCMC sampling ideas and principles are discussed in detail. Fitting selected models through MCMC sampling is illustrated using simple synthetic data sets, with example R code available in a Github repository.