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Entropy 2014, 16(6), 3074-3102; doi:10.3390/e16063074
Article

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

1,*  and 2
1 Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, UK 2 Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
* Author to whom correspondence should be addressed.
Received: 29 March 2014 / Revised: 23 May 2014 / Accepted: 28 May 2014 / Published: 3 June 2014
(This article belongs to the Special Issue Information Geometry)
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Abstract

Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this, geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.
Keywords: information geometry; Markov chain Monte Carlo; Bayesian inference; computational statistics; machine learning; statistical mechanics; diffusions information geometry; Markov chain Monte Carlo; Bayesian inference; computational statistics; machine learning; statistical mechanics; diffusions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Livingstone, S.; Girolami, M. Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions. Entropy 2014, 16, 3074-3102.

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