Case Studies in Bayesian Statistical Modelling and Analysis

16

Bayesian Mixture Models: When the Thing you need to know is the Thing you cannot Measure

Clair L. Alston¹, Kerrie L. Mengersen¹ and Graham E. Gardner²

¹Queensland University of Technology, Brisbane, Australia

²Murdoch University, Perth, Australia

16.1 Introduction

There are many situations in which the data analyst may find finite mixture models helpful. In general, they are used when it is thought that there may be more than one population present in the data set, and either by oversight, or more usually as a result of being unable to identify subpopulations, the data may contain a mixture of measurements from several populations. Alternatively, they may be used in a semi-parametric density estimation setting.

This concept was first expounded in Pearson (1894), who fitted a mixture of two normal distributions to a data set which contained the forehead to body length ratio measurements of sampled crabs residing in the Bay of Naples. The skewness of the data prompted speculation of species evolution delivering two subpopulations, with mixtures being used as a confirmation (McLachlan and Peel 2000, pp. 2–3).

The method of moments proposed by Pearson (1894) was computationally demanding and as a result mixture modelling was rarely used for data analysis until Dempster et al. (1977) produced their seminal paper using the EM algorithm, and illustrated its ability to simplify the application of maximum likelihood techniques to incomplete data settings. This approach had a direct ...