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.