Sequentially Interacting Markov Chain Monte Carlo Methods

We introduce a novel methodology for sampling from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. These problems are usually addressed using Sequential Monte Carlo (SMC) methods. The alternative Sequentially Interacting Markov Chain Monte Carlo (SIMCMC) scheme proposed here works by generating interacting non-Markovian sequences which behave asymptotically like independent Metropolis-Hastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC methods, this scheme allows us to iteratively improve our estimates in an MCMC-like fashion. We establish convergence of the algorithm under realistic verifiable assumptions and demonstrate its performance on several examples arising in Bayesian time series analysis.