Information-geometric Markov Chain Monte Carlo methods using Diffusions

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 appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.

22 pages, 2 figures

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• The Random Walk Metropolis: Linking Theory and Practice Through a Case Study
• Langevin diffusions and the Metropolis-adjusted Langevin algorithm
• Geodesic Monte Carlo on Embedded Manifolds
• Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets
• Proposals which speed-up function-space MCMC
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Cited by in corpus (3)

• On the Geometric Ergodicity of Hamiltonian Monte Carlo
• Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents