Asymptotic Analysis of the Random-Walk Metropolis Algorithm on Ridged Densities

Abstract: In this paper we study the asymptotic behavior of the Random-Walk Metropolis algorithm on probability densities with two different scales', where most of the probability mass is distributed along certain key directions with theorthogonal' directions containing relatively less mass. Such class of probability measures arise in various applied contexts including Bayesian inverse problems where the posterior measure concentrates on a sub-manifold when the noise variance goes to zero. When the target measure concentrates on a linear sub-manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting Stochastic Differential Equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).