On the Optimal Scaling of the Modified Metropolis-Hastings algorithm

ABSTRACT: Estimation of small failure probabilities is one of the most important and challenging problems in reliability engineering. In cases of practical interest, the failure probability is given by a high-dimensional integral. Since multivariate integration suffers from the curse of dimensionality, the usual numerical methods are inapplicable. Over the past decade, the civil engineering research community has increasingly realized the potential of advanced simulation methods for treating reliability problems. The Subset Simulation method, in-troduced by Au & Beck (2001a), is considered to be one of the most robust advanced simulation techniques for solving high-dimensional nonlinear problems. The Modified Metropolis-Hastings (MMH) algorithm, a varia-tion of the original Metropolis-Hastings algorithm (Metropolis et al. 1953, Hastings 1970), is used in Subset Simulation for sampling from conditional high-dimensional distributions. The efficiency and accuracy of Subset Simulation directly depends on the ergodic properties of the Markov chain generated by MMH, in other words, on how fast the chain explores the parameter space. The latter is determined by the choice of one-dimensional proposal distributions, making this choice very important. It was noticed in Au & Beck (2001a) that the perfor-mance of MMH is not sensitive to the type of the proposal PDFs, however, it strongly depends on the variance of proposal PDFs. Nevertheless, in almost all real-life applications, the scaling of proposal PDFs is still largely an art. The issue of optimal scaling was realized in the original paper by Metropolis (Metropolis et al. 1953)