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Markov chain Monte Carlo on Riemann manifolds

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If you have a question about this talk, please contact Richard Nickl.

Markov chain Monte Carlo (MCMC) provides the dominant methodology for inference over statistical models with non-conjugate priors. Despite a wealth of theoretical characterisation of mixing times, geometric ergodicity, and asymptotic step-sizes, the design and implementation of MCMC methods remains something of an engineering art-form. An attempt to address this issue in a systematic manner leads one to consider the geometry of probability distributions, as has been the case previously in the study of e.g. higher-order efficiency in statistical estimators. By considering the natural Riemannian geometry of probability distributions MCMC proposal mechanisms based on Langevin diffusions that are characterised by the metric tensor and associated manifold connections are proposed and studied. Furthermore, optimal proposals that follow the geodesic paths related to the metric are defined via the Hamilton-Jacobi approach and these are empirically evaluated on some challenging modern-day inference tasks.

This talk is based on work that was presented as a Discussion Paper to the Royal Statistical Society and a dedicated website with Matlab codes is available at

This talk is part of the Statistics series.

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