  Because of their multimodality, mixture posterior distributions are difficult
to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a
strategy to enhance the sampling of MCMC in this context, using a biasing
procedure which originates from computational Statistical Physics. The
principle is first to choose a "reaction coordinate", that is, a "direction" in
which the target distribution is multimodal. In a second step, the marginal
log-density of the reaction coordinate with respect to the posterior
distribution is estimated; minus this quantity is called "free energy" in the
computational Statistical Physics literature. To this end, we use adaptive
biasing Markov chain algorithms which adapt their targeted invariant
distribution on the fly, in order to overcome sampling barriers along the
chosen reaction coordinate. Finally, we perform an importance sampling step in
order to remove the bias and recover the true posterior. The efficiency factor
of the importance sampling step can easily be estimated \emph{a priori} once
the bias is known, and appears to be rather large for the test cases we
considered. A crucial point is the choice of the reaction coordinate. One
standard choice (used for example in the classical Wang-Landau algorithm) is
minus the log-posterior density. We discuss other choices. We show in
particular that the hyper-parameter that determines the order of magnitude of
the variance of each component is both a convenient and an efficient reaction
coordinate. We also show how to adapt the method to compute the evidence
(marginal likelihood) of a mixture model. We illustrate our approach by
analyzing two real data sets.
