Chapter

Free Energy Sequential Monte Carlo, Application to Mixture Modelling*

Nicolas Chopin and Pierre Jacob

in Bayesian Statistics 9

Published in print October 2011 | ISBN: 9780199694587
Published online January 2012 | e-ISBN: 9780191731921 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199694587.003.0003
Free Energy Sequential Monte Carlo, Application to Mixture Modelling*

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We introduce a new class of Sequential Monte Carlo (SMC) methods, which we call free energy SMC. This class is inspired by free energy methods, which originate from physics, and where one samples from a biased distribution such that a given function ξ(θ) of the state θ is forced to be uniformly distributed over a given interval. From an initial sequence of distributions (π t) of interest, and a particular choice of ξ(θ), a free energy SMC sampler computes sequentially a sequence of biased distributions (π̃t) with the following properties: (a) the marginal distribution of ξ(θ) with respect to π̃t is approximatively uniform over a specified interval, and (b) π̃t and π t have the same conditional distribution with respect to ξ. We apply our methodology to mixture posterior distributions, which are highly multimodal. In the mixture context, forcing certain hyper‐parameters to higher values greatly facilitates mode swapping, and makes it possible to recover a symmetric output. We illustrate our approach with univariate and bivariate Gaussian mixtures and two real‐world datasets.

Keywords: Free energy biasing; Label switching; Mixture; Sequential Monte Carlo; particle filter

Chapter.  12888 words.  Illustrated.

Subjects: Probability and Statistics

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