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Bayesian Statistics 9$
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José M. Bernardo, M. J. Bayarri, James O. Berger, A. P. Dawid, David Heckerman, Adrian F. M. Smith, and Mike West

Print publication date: 2011

Print ISBN-13: 9780199694587

Published to Oxford Scholarship Online: January 2012

DOI: 10.1093/acprof:oso/9780199694587.001.0001

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Free Energy Sequential Monte Carlo, Application to Mixture Modelling *

Free Energy Sequential Monte Carlo, Application to Mixture Modelling *

Chapter:
(p.91) Free Energy Sequential Monte Carlo, Application to Mixture Modelling*
Source:
Bayesian Statistics 9
Author(s):

Nicolas Chopin

Pierre Jacob

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199694587.003.0003

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 (π̃ t ) with the following properties: (a) the marginal distribution of ξ(θ) with respect to π̃ t is approximatively uniform over a specified interval, and (b) π̃ t 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

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