Modified Metropolis–Hastings algorithm with delayed rejection

Abstract

The development of an efficient MCMC strategy for sampling from complex distributions is a difficult task that needs to be solved for calculating the small failure probabilities encountered in the high-dimensional reliability analysis of engineering systems. Usually different variations of the Metropolis–Hastings algorithm (MH) are used. However, the standard MH algorithm does not generally work in high dimensions, since it leads to very frequent repeated samples. In order to overcome this deficiency one can use the Modified Metropolis–Hastings algorithm (MMH) proposed in Au and Beck (2001) [1]. Another variation of the MH algorithm, called the Metropolis–Hastings algorithm with delayed rejection (MHDR) has been proposed by Tierney and Mira (1999) [7]. The key idea behind the MHDR algorithm is to reduce the correlation between states of the Markov chain. In this paper we combine the ideas of MMH and MHDR and propose a novel modification of the MH algorithm, called the Modified Metropolis–Hastings algorithm with delayed rejection (MMHDR). The efficiency of the new algorithm is demonstrated with a numerical example where MMHDR is used together with Subset simulation for computing small failure probabilities in high dimensions.