Interacting sequential Monte Carlo samplers for trans-dimensional simulation

Abstract

The methodology of interacting sequential Monte Carlo (SMC) samplers is introduced. SMC samplers are methods for sampling from a sequence of densities on a common measurable space using a combination of Markov chain Monte Carlo (MCMC) and sequential importance sampling/resampling (SIR) methodology. One of the main problems with SMC samplers when simulating from trans-dimensional, multimodal static targets is that transition kernels do not mix which leads to low particle diversity. In such situations poor Monte Carlo estimates may be derived. To deal with this problem an interacting SMC approach for static inference is introduced. The method proceeds by running SMC samplers in parallel on, initially, different regions of the state space and then moving the corresponding samples onto the entire state space. Once the samplers reach a common space the samplers are combined and allowed to interact. The method is intended to increase the diversity of the population of samples. It is established that interacting SMC admit a Feynman–Kac representation; this provides a framework for the convergence results that are developed. In addition, the methodology is demonstrated on a trans-dimensional inference problem in Bayesian mixture modelling and also, using adaptive methods, a mixture modelling problem in population genetics.