Variational Bayes approach for model aggregation in unsupervised classification with Markovian dependency

A binary unsupervised classification problem where each observation is associated with an unobserved label that needs to be retrieved is considered. More precisely, it is assumed that there are two groups of observation: normal and abnormal. The ‘normal’ observations are coming from a known distribution whereas the distribution of the ‘abnormal’ observations is unknown. Several models have been developed to fit this unknown distribution. An alternative based on a mixture of Gaussian distributions is proposed. The inference is performed within a variational Bayesian framework and the aim is to infer the posterior probability of belonging to the class of interest. To this end, it makes little sense to estimate the number of mixture components since each mixture model provides more or less relevant information to the posterior probability estimation. By computing a weighted average (named aggregated estimator) over the model collection, Bayesian Model Averaging (BMA) is one way of combining models in order to account for information provided by each model. An aim is then the estimation of the weights and the posterior probability for a specific model. Optimal approximations of these quantities from the variational theory are derived; other approximations of the weights are also proposed. It is assumed that the data are dependent (Markovian dependency) and hence a Hidden Markov Model is considered. A simulation study is carried out to evaluate the accuracy of the estimates in terms of classification performance. An illustration on both epidemiologic and genetic datasets is presented.