Roberts, G. O.; Gelman, A.; Gilks, W. R. Weak convergence and optimal scaling of random walk Metropolis algorithms. (English) Zbl 0876.60015 Ann. Appl. Probab. 7, No. 1, 110-120 (1997). Summary: 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. Cited in 6 ReviewsCited in 315 Documents MSC: 60F05 Central limit and other weak theorems 65C99 Probabilistic methods, stochastic differential equations Keywords:Metropolis algorithm; weak convergence; optimal scaling; Markov chain Monte Carlo PDFBibTeX XMLCite \textit{G. O. Roberts} et al., Ann. Appl. Probab. 7, No. 1, 110--120 (1997; Zbl 0876.60015) Full Text: DOI