Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

TL;DR

This paper analyzes the optimal scaling of high-dimensional random walk Metropolis algorithms with discontinuous target densities, establishing convergence to Langevin diffusions and identifying an optimal acceptance rate of approximately 13.5%.

Contribution

It extends the optimal scaling theory to discontinuous target densities, providing new convergence results and practical guidelines for algorithm efficiency.

Findings

Optimal acceptance rate of about 13.5%.

Convergence to Langevin diffusion as dimension increases.

Highlights limitations of RWM over Metropolis-within-Gibbs.

Abstract

We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to infinity. In particular, when the proposal variance is scaled by $d^{-2}$, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of $e^{-2}$ (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.