## The Annals of Applied Probability

### Weak convergence and optimal scaling of random walk Metropolis algorithms

#### Abstract

This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to $\infty$. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process.

The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions.

The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.

#### Article information

Source
Ann. Appl. Probab. Volume 7, Number 1 (1997), 110-120.

Dates
First available in Project Euclid: 14 October 2002

http://projecteuclid.org/euclid.aoap/1034625254

Digital Object Identifier
doi:10.1214/aoap/1034625254

Mathematical Reviews number (MathSciNet)
MR1428751

Zentralblatt MATH identifier
0876.60015

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 65U05

#### Citation

Roberts, G. O.; Gelman, A.; Gilks, W. R. Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 (1997), no. 1, 110--120. doi:10.1214/aoap/1034625254. http://projecteuclid.org/euclid.aoap/1034625254.