13 - Colouring and breaking sticks: random distributions and heterogeneous clustering

Summary

Abstract

We begin by reviewing some probabilistic results about the Dirichlet Process and its close relatives, focussing on their implications for statistical modelling and analysis. We then introduce a class of simple mixture models in which clusters are of different ‘colours’, with statistical characteristics that are constant within colours, but different between colours. Thus cluster identities are exchangeable only within colours. The basic form of our model is a variant on the familiar Dirichlet process, and we find that much of the standard modelling and computational machinery associated with the Dirichlet process may be readily adapted to our generalisation. The methodology is illustrated with an application to the partially-parametric clustering of gene expression profiles.

Keywords Bayesian nonparametrics, gene expression profiles, hierarchical models, loss functions, MCMC samplers, optimal clustering, partition models, Pólya urn, stick breaking

AMS subject classification (MSC2010) 60G09, 62F15, 62G99, 62H30, 62M99

Introduction

The purpose of this note is four-fold: to remind some Bayesian nonparametricians gently that closer study of some probabilistic literature might be rewarded, to encourage probabilists to think that there are statistical modelling problems worth of their attention, to point out to all another important connection between the work of John Kingman and modern statistical methodology (the role of the coalescent in population genetics approaches to statistical genomics being the most important example; see papers by Donnelly, Ewens and Griffiths in this volume), and finally to introduce a modest generalisation of the Dirichlet process.