Interacting sequential Monte Carlo samplers for trans-dimensional simulation

Authors:
Ajay Jasra;Arnaud Doucet;David A. Stephens;Christopher C. Holmes
Affiliations:
Department of Mathematics, Imperial College London, SW7 2AZ, UK;Departments of Statistics & Computer Science, University of British Columbia, Canada V6T 1Z2;Departments of Mathematics & Statistics, McGill University, Canada H3A 2K6;Department of Statistics, University of Oxford, OX1 3TG, UK
Venue:
Computational Statistics & Data Analysis
Year:
2008

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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.