Reversible jump MCMC

Statistical problems where ‘the number of things you don’t know is one of the things you don’t know’ are ubiquitous in statistical modelling. They arise both in traditional modelling situations such as variable selection in regression, and in more novel methodologies such as object recognition, signal processing, and Bayesian nonparametrics. All such ‘trans-dimensional’ problems can be formulated generically, sometimes with a little ingenuity, as a matter of joint inference about a model indicator k and a parameter vector θk, where the model indicator determines the dimension nk of the parameter, but this dimension varies from model to model. Almost invariably in a frequentist setting, inference about these two kinds of unknown is based on different logical principles, but, at least formally, the Bayes paradigm offers the opportunity of a single logical framework – it is the joint posterior π(k, θk|Y ) of model indicator and parameter given data Y that is the basis for inference. Reversible jump Markov chain Monte Carlo (Green, 1995) is a method for computing this posterior distribution by simulation, or more generally, for simulating from a Markov chain whose state is a vector whose dimension is not fixed. It has many applications other than in Bayesian statistics. Much of what follows will apply equally to them all; however, for simplicity, we will use the Bayesian motivation and terminology throughout. The joint inference problem can be set naturally in the form of a simple Bayesian hierarchical model. We suppose that a prior p(k) is specified over models k in a countable set K, and for each k we are given a prior distribution p(θk|k), along with a likelihood L(Y |k, θk) for the data Y . For simplicity of exposition, we suppose that p(θk|k) is a probability density, and that there are no other parameters, so that where there are parameters common to all models these are subsumed into each θk ∈ Xk ⊂ Rnk . Additional parameters, perhaps in additional layers of a hierarchy, are easily dealt with. Note that in this chapter, all probability distributions are proper. In some settings, p(k) and p(θk|k) are not separately available, even up to multiplicative constants; this applies for example in many point process models. However it will be clear that what follows requires specification only of the product p(k, θk) = p(k)×p(θk|k) of these factors, up to a multiplicative constant. In many models there are discrete unknowns as well as continuously distributed ones. Such unknowns, whether fixed or variable in number, cause no additional difficulties; only discrete-state Markov chain notions are needed to