  Mixture models are widely used in Bayesian statistics and machine learning,
in particular in computational biology, natural language processing and many
other fields. Variational inference, a technique for approximating intractable
posteriors thanks to optimization algorithms, is extremely popular in practice
when dealing with complex models such as mixtures. The contribution of this
paper is two-fold. First, we study the concentration of variational
approximations of posteriors, which is still an open problem for general
mixtures, and we derive consistency and rates of convergence. We also tackle
the problem of model selection for the number of components: we study the
approach already used in practice, which consists in maximizing a numerical
criterion (the Evidence Lower Bound). We prove that this strategy indeed leads
to strong oracle inequalities. We illustrate our theoretical results by
applications to Gaussian and multinomial mixtures.
