B0621
Title: Explicit convergence bounds for Metropolis Markov chains
Authors: Samuel Power - University of Bristol (United Kingdom) [presenting]
Abstract: Markov chain Monte Carlo (MCMC) algorithms are a widely-used tool for approximate simulation from probability measures in structured, high-dimensional spaces, with a variety of applications. A key ingredient of their success is their ability to converge rapidly to equilibrium at a rate which depends acceptably on the difficulty of the sampling problem at hand, as captured by the dimension of the problem, and the concentration and smoothness properties of the target distribution. The objective is to present the convergence analysis of Metropolis-type MCMC algorithms on Euclidean spaces. In particular, a detailed study of the random walk Metropolis (RWM) Markov chain is provided with arbitrary proposal variances and in any dimension, obtaining interpretable estimates on their convergence behaviour under suitable assumptions. These estimates have a provably sharp dependence on the dimension of the problem, thus providing theoretical validation for the use of these algorithms in complex settings. The positive results are quite generally applicable. The preconditioned Crank-Nicolson Markov chain is studied as applied to simulation from Gaussian Process posterior models, obtaining dimension-independent complexity bounds under suitable assumptions.
