ABSTRACT
This chapter describes the basic ideas of the subject, present several alternative representations and perspectives on these models, and discusses some of the elements of inference about the unknowns in the models. It focuses on the simplest set-up, of finite mixture models and also discusses how various simplifying assumptions can be relaxed to generate the rich landscape of modelling and inference ideas. The kind of countably infinite mixture that has been most important in applications is the Dirichlet process mixture, and its relatives, that form a central methodology in Bayesian nonparametric modelling. Mixture models have been around for over 150 years, as an intuitively simple and practical tool for enriching the collection of probability distributions available for modelling data. Where data are indexed spatially rather than temporally, mixture models can play an analogous role, and as usual the models proposed can often be viewed as generalizations or translations from time to space.
