ABSTRACT
The reversible jump Markov chain Monte Carlo (MCMC) sampler (Green, 1995) provides a general framework for Markov chain Monte Carlo simulation in which the dimension of the parameter space can vary between iterates of the Markov chain. The reversible jump sampler can be viewed as an extension of the Metropolis-Hastings algorithm onto more general state spaces. To understand this in a Bayesian modeling context, suppose that for observed data x we
havea countable collectionof candidatemodelsM = {M1,M2, . . .} indexedbyaparameter k ∈ K. The index k can be considered as an auxiliarymodel indicator variable, such thatMk′ denotes themodel where k = k′. EachmodelMk has an nk-dimensional vector of unknown parameters, θk ∈ Rnk , where nk can take different values for different models k ∈ K. The joint posterior distribution of (k, θk) given observed data, x, is obtained as the product of the likelihood, L(x | k, θk), and the joint prior, p(k, θk) = p(θk | k)p(k), constructed from the prior distribution of θk under model Mk , and the prior for the model indicator k (i.e. the prior for model Mk). Hence, the joint posterior is
π(k, θk | x) = L(x | k, θk)p(θk | k)p(k)∑ k′∈K
∫ R nk′ L(x | k′, θ′k′)p(θ′k′ | k′)p(k′)dθ′k′
. (3.1)
The reversible jump algorithm uses the joint posterior distribution in Equation 3.1 as the target of an MCMC sampler over the state space Θ = ⋃k∈K({k} × Rnk ), where the states of the Markov chain are of the form (k, θk), the dimension of which can vary over the state space. Accordingly, from the output of a single Markov chain sampler, the user is able to obtain a full probabilistic description of the posterior probabilities of each model having observed the data, x, in addition to the posterior distributions of the individual models. This chapter aims to provide an overview of the reversible jump sampler. Wewill outline
the sampler’s theoretical underpinnings, present the latest and most popular techniques for enhancing algorithmperformance, and discuss the analysis of sampler output. Through the use of numerous worked examples it is hoped that the reader will gain a broad appreciation of the issues involved in multi-model simulation, and the confidence to implement reversible jump samplers in the course of their own studies.
