Abstract

The Random Walk Metropolis (RWM) is a simple and enduring Markov chain-based algorithm for approximate simulation from an intractable ‘target’ probability distribution. In a pair of recent works, we have undertaken a detailed study of the quantitative convergence of this algorithm to its equilibrium distribution, establishing non-asymptotic estimates on mixing times, with explicit dependence on dimension and other relevant problem parameters. The results hold at a reasonable level of generality, and are often sharp in a suitable sense. The focus of the talk will be conceptual rather than technical, with an eye towards enabling intuition for i) which high-level aspects of the target distribution influence the convergence behaviour of RWM , and ii) which concrete properties must be verified in order to obtain a rigorous proof. A key element will be the impact of tail behaviour and measure concentration on the convergence profile of the algorithm across different time-scales.  No prior knowledge of the RWM is required from the audience. (joint work with C. Andrieu, A. Lee and A. Wang)