Optimal Scaling of Random Walk Metropolis algorithms with\nDiscontinuous target densities

Abstract

We consider the optimal scaling problem for high-dimensional\nRandom walk Metropolis (RWM) algorithms where the target\ndistribution has a discontinuous probability density function. All\nprevious analysis has focused upon continuous target densities.\nThe main result is a weak convergence result as the dimensionality\n$d$ of the target densities converges to $\\infty$. In particular,\nwhen the proposal variance is scaled by $d^{-2}$, the sequence of\nstochastic processes formed by the first component of each Markov\nchain converges to an appropriate Langevin diffusion process.\nTherefore optimising the efficiency of the RWM algorithm is\nequivalent to maximising the speed of the limiting diffusion. This\nleads to an asymptotic optimal acceptance rate of $e^{-2}\n(=0.1353)$ under quite general conditions. The results have major\npractical implications for the implementation of RWM algorithms by\nhighlighting the detrimental effect of choosing RWM algorithms\nover Metropolis-within-Gibbs algorithms.