Combinatorial Mixtures of Multiparameter Distributions: an Application to Prostate Cancer

The term combinatorial mixtures refers to a flexible class of parametric models for inference on mixture distributions whose components have multidimensional parameters [1]. The idea behind it is to allow each element of the component-specific parameter vector to be shared by a subset of other components. We develop Bayesian inference and computational approaches based on Markov Chain Monte Carlo methods for this class of mixture distributions with an unknown number of components. We define the structure for a general prior distribution - a mixture of prior distributions itself - where a positive probability is put on every possible combination of sharing patterns. We illustrate our approach in an application based on the normal mixture model for bivariate data. We assume a decomposition of the covariance matrix which allows to model standard deviations and correlations separately. We also discuss solutions to the 'label switching' problem. For our application, we use publicly available data on mRNA expression in prostate carcinoma [2], where a two-component 'ellipsoidal, varying volume, shape, and orientation' model has been suggested by a different approach [3].