Following a moving targetÐMonte Carlo inference for dynamic Bayesian models Walter R. Gilks Medical Research Council Biostatistics Unit, Cambridge, UK and Carlo Berzuini University of Pavia, Italy [Received September 1998. Final revision August 2000] Summary. Markov chain Monte Carlo MCMC) sampling is a numerically intensive simulation technique which has greatly improved the practicality of Bayesian inference and prediction. How- ever, MCMC sampling is too slow to be of practical use in problems involving a large number of posterior target) distributions, as in dynamic modelling and predictive model selection. Alternative simulation techniques for tracking moving target distributions, known as particle ®lters, which combine importance sampling, importance resampling and MCMC sampling, tend to suffer from a progressive degeneration as the target sequence evolves. We propose a new technique, based on these same simulation methodologies, which does not suffer from this progressive degeneration. Keywords: Bayesian inference; Dynamic model; Hidden Markov model; Importance resampling; Importance sampling; Markov chain Monte Carlo methods; Particle ®lter; Predictive model selection; Sequential imputation; Simulation; Tracking 1. Introduction Bayesian applications of Markov chain Monte Carlo MCMC) methods involve generating many samples from the posterior distribution of the model parameters by using a Markov chain, and then approximating posterior expectations of interest with sample averages. Although MCMC sampling is computationally intensive, it has been remarkably successful in expanding the repertoire of feasible Bayesian problems; see, for example, the many applica- tions described in Gilks et al. 1996). However, for dynamic problems where the posterior or target) distribution evolves over time through the accumulation of data, and in other situations where a large collection of target distributions is involved, MCMC methods are too computationally intensive to be useful, especially where realtime sequential forecasting is required. Examples of dynamic problems include ®nancial and medical time series prediction, sequential system identi®cation in control engineering, speech recognition, military tracking, on-line updating of classi®cation systems and machine learning. Multiple-target distributions also arise with model selection techniques based on k-step-ahead prediction. To reduce the computational burden of dynamic Bayesian analysis, several techniques involving some or all of importance sampling, importance resampling and MCMC sampling have been proposed Handschin and Mayne, 1969; Zaritskii et al., 1975; Kong et al., 1994; Address for correspondence: Walter R. Gilks, Medical Research Council Biostatistics Unit, Institute of Public Health, University Forvie Site, Robinson Way, Cambridge, CB2 2SR, UK. E-mail:wally.gilks@mrc-bsu.cam.ac.uk & 2001 Royal Statistical Society 1369±7412/01/63127 J. R. Statist. Soc. B 2001) 63, Part 1, pp.127±146