Tatiana Xifara (University of California, Santa Cruz): Diffusions with position dependent volatility and the Metropolis adjusted Langevin algorithm

In this poster, we provide a clarification of the description of Langevin diffusions on Riemannian manifolds and of the measure underlying the invariant density. As a result we propose a new position-dependent Metropolis-adjusted Langevin algorithm (PMALA) based upon a Langevin diffusion in Rd which has the required invariant density with respect to Lebesgue measure. We compare analytically the two algorithms, i.e. the one based on our diffusion and the previously-proposed position-dependent MALA (MMALA) in Girolami \& Calderhead (2011)) and show that they are equivalent in some cases but are distinct in general. We also correct a transcription error in the definition of the Langevin diffusions with invariant density with respect to Hausdorff (or volume) measure that MMALA is based on. A simulation study illustrates the gain in efficiency provided by the new position-dependent MALA.

Langevin diffusions

Metropolis adjusted Langevin algorithm
Riemann manifolds

Chris Sherlock, Sam Livingstone, Simon Byrne, Mark Girolami