## The Annals of Applied Probability

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

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

A. Gelman, W. R. Gilks, and G. O. Roberts

#### 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

**Permanent link to this document**

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

**Keywords**

Metropolis algorithm weak convergence optimal scaling Markov chain Monte Carlo

#### 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.