## The Annals of Applied Probability

### Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

#### 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 $\infty$. 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.

#### Article information

Source
Ann. Appl. Probab., Volume 22, Number 5 (2012), 1880-1927.

Dates
First available in Project Euclid: 12 October 2012

https://projecteuclid.org/euclid.aoap/1350067989

Digital Object Identifier
doi:10.1214/11-AAP817

Mathematical Reviews number (MathSciNet)
MR3025684

Zentralblatt MATH identifier
1259.60082

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 65C05: Monte Carlo methods

#### Citation

Neal, Peter; Roberts, Gareth; Yuen, Wai Kong. Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Ann. Appl. Probab. 22 (2012), no. 5, 1880--1927. doi:10.1214/11-AAP817. https://projecteuclid.org/euclid.aoap/1350067989

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