# Free Energy Methods for Efficient Exploration of Mixture Posterior Densities

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## Abstract

Because of their multimodality, mixture posterior densities are difficult to sample withstandard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhancethe sampling of MCMC in this context, using a biasing procedure which originates fromcomputational statistical physics. The principle is first to choose a “reaction coordinate”,that is, a direction in which the target density is multimodal. In a second step, the marginallog-density of the reaction coordinate is estimated; this quantity is called “free energy” inthe computational statistical physics literature. To this end, we use adaptive biasing Markovchain algorithms which adapt their invariant distribution on the fly, in order to overcomesampling barriers along the chosen reaction coordinate. Finally, we perform an importancesampling step in order to remove the bias and recover the true posterior. The efficiency factorcan easily be estimated a priori once the bias is known, and is large enough for the test caseswe considered.A crucial point is the choice of the reaction coordinate. One standard choice (used forexample in the classical Wang-Landau algorithm) is the opposite of the log-posterior density.We show that another convenient and efficient reaction coordinate is the hyper-parameterthat determines the order of magnitude of the variance of each component. We also showhow to adapt the method to perform model choice between different number of components.We illustrate our approach by analyzing two real data sets.

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