  We consider a binary unsupervised classification problem where each
observation is associated with an unobserved label that we want to retrieve.
More precisely, we assume 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. In this paper, we
propose an alternative based on a mixture of Gaussian distributions. The
inference is done within a variational Bayesian framework and our aim is to
infer the posterior probability of belonging to the class of interest. To this
end, it makes no sense to estimate the mixture component number 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.
The aim is then the estimation of the weights and the posterior probability for
one specific model. In this work, we derive optimal approximations of these
quantities from the variational theory and propose other approximations of the
weights. To perform our method, we consider that the data are dependent
(Markovian dependency) and hence we consider a Hidden Markov Model. A
simulation study is carried out to evaluate the accuracy of the estimates in
terms of classification. We also present an application to the analysis of
public health surveillance systems.
