 2003 Royal Statistical Society 1369–7412/03/65003 J. R. Statist. Soc. B (2003) 65, Part 1, pp. 3–55 Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions S. P. Brooks, University of Cambridge, UK P. Giudici University of Pavia, Italy and G. O. Roberts Lancaster University, UK [Read before The Royal Statistical Society at a meeting organized by the Research Section at the 65th Annual Meeting of the Institute of Mathematical Statistics in Banff on Tuesday, July 30th, 2002 , Professor B. W. Silverman in the Chair ] Summary. The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of accep- tance probabilities for jumps.Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms.The second group of methods generalizes the reversible jump algorithm by using the so-called saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed.The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illus- trate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling. Keywords: Autoregressive time series; Bayesian model selection; Graphical models; Langevin algorithms; Mixture modelling; Optimal scaling 1. Introduction Thereversiblejumpalgorithm(Green,1995)isanextensionofthepopularMetropolis–Hastings algorithm,designedtoallowmovementbetweendifferentdimensionalspaces.Thesealgorithms aremostcommonlyappliedto(Bayesian)modeldeterminationproblems(DellaportasandFor- ster,1999;RichardsonandGreen,1997;FanandBrooks,2000)thoughotherapplicationsexist (e.g. Møller (1999) and Brooks et al. (2003)). We shall focus on the Bayesian model determi- nation problem here and consider issues such as the choice of prior and specification of the likelihoodasbeyondthescopeofthepaper.Thus,weareconcernedsolelywithusingreversible Address for correspondence: S.P.Brooks,StatisticalLaboratory,UniversityofCambridge,WilberforceRoad, Cambridge,CB30WB,UK. E-mail: steve@statslab.cam.ac.uk