Sam Livingstone (University College London): A Tutorial on diffussions on manifolds for MCMC

Some recent research in Markov Chain Monte Carlo has focussed on generalising existing algorithms that unfold on Euclidean space to an arbitrary Riemannian manifold (e.g. Girolami & Calderhead, 2011). If the state space of a Markov chain is viewed as some manifold which is homeomorphic to \mathbb{R}^n, rather than simply \mathbb{R}^n itself, then distances can be warped, so that Markov chain samplers can reach to the typical set more speedily (improved convergence) and explore it more efficiently once there (better mixing). This poster will show from first principles how to map a diffusion onto a manifold, as referenced in (Xifara et. al., 2013), and show the connections between viewing the space as a manifold and pre-conditioning a method. Although the flavour is expository the results are novel and clarify some definitions which have previously been unclear among statisticians. This foundational work reveals a new Markov chain Monte Carlo method based on Langevin diffusions (Xifara et. al., 2013), the empirical performance of which will be shown in the poster of T. Xifara.

References:

Girolami, Mark, and Ben Calderhead. “Riemann manifold Langevin and Hamiltonian Monte Carlo methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73.2 (2011): 123-214.

Xifara, Tatiana, et al. “Langevin diffusions and the Metropolis-adjusted Langevin algorithm.” arXiv preprint arXiv:1309.2983 (2013).

Roberts, Gareth O., and Osnat Stramer. “Langevin diffusions and Metropolis-Hastings algorithms.” Methodology and computing in applied probability 4.4 (2002): 337-357.

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