Metropolis based MCMC methods are an effective and flexible tool for sampling a wide variety of complex probability distributions. Nonetheless, their effective use depends very much on careful tuning of parameters, choice of proposal distribution and so forth. A thorough understanding of these issues in high dimensional problems is particularly desirable as they can be critical to the construction of a practical sampler.
In this talk we confine ourselves to proposal distributions based on simple random walks, and on discretizations of the Langevin equation. We review previous work by Roberts and co-workers which concerns target distributions with a product structure. We then show how ideas from numerical analysis can be used to construct improved samplers for such problems, quantifying these assertions analytically, and illustrating them numerically. Throughout we illustrate with application to the sampling of conditioned diffusion processes, an applied field in which the ideas presented have significant practical impact.
Collaboration with Alex Beskos (Warwick), Gareth Roberts (Lancaster) and Jochen Voss (Warwick).