Unsupervised Learning of Correlated Multivariate Gaussian Mixture Models Using MML

Abstract

Mixture modelling or unsupervised classification is the problem of identifying and modelling components (or clusters, or classes) in a body of data. We consider here the application of the Minimum Message Length (MML) principle to a mixture modelling problem of multivariate Gaussian distributions. Earlier work in MML mixture modelling includes the multinomial, Gaussian, Poisson, von Mises circular, and Student t distributions and in these applications all variables in a component are assumed to be uncorrelated with each other. In this paper, we propose a more general type of MML mixture modelling which allows the variables within a component to be correlated. Two MML approximations are used. These are the Wallace and Freeman (1987) approximation and Dowe’s MMLD approximation (2002). The former is used for calculating the relative abundances (mixing proportions) of each component and the latter is used for estimating the distribution parameters involved in the components of the mixture model. The proposed method is applied to the analysis of two real-world datasets – the well-known (Fisher) Iris and diabetes datasets. The modelling results are then compared with those obtained using two other modelling criteria, AIC and BIC (which is identical to Rissanen’s 1978 MDL), in terms of their probability bit-costings, and show that the proposed MML method performs better than both these criteria. Furthermore, the MML method also infers more closely the three underlying Iris species than both AIC and BIC.