On MCMC algorithm for Subset Simulation

A new Markov Chain Monte Carlo (MCMC) algorithm for Subset Simulation was recently proposed by imposing a joint Gaussian distribution between the current sample and the candidate. It coincides with the limiting case of the original independent-component algorithm where each random variable is represented by an infinite number of hidden variables. The algorithm is remarkably simple as it no longer involves the explicit choice of proposal distribution. It opens up a new perspective for generating conditional failure samples and potentially allows more direct and flexible control of algorithm through the cross correlation matrix between the current sample and the candidate. While by definition the cross correlation matrix need not be symmetric, this article shows that it must be so in order to satisfy detailed balance and hence to produce an unbiased algorithm. The effect of violating symmetry on the distribution of samples is discussed and insights on acceptance probability are provided.