RJMCMC

RJMCMC, short for Reversible Jump Markov Chain Monte Carlo, is a class of Markov chain Monte Carlo methods that extends the Metropolis-Hastings framework to parameter spaces of varying dimension. It enables joint inference over a set of competing models and their parameters, allowing Bayesian model determination and calculation of posterior model probabilities. It was introduced by Peter Green in 1995.

The central idea is to construct reversible moves between models with different numbers of parameters by introducing auxiliary variables and a dimension-matching transformation so that the move is bijective. Each proposed move from model M with parameter vector theta to model M' with theta' uses a proposal involving an auxiliary variable u; theta', u are drawn from a proposal distribution, and the Jacobian determinant of the transformation is incorporated in the acceptance probability. This ensures the chain satisfies detailed balance with respect to the joint posterior over models and parameters.

Common move types include birth or death moves that add or remove parameters (for example adding or removing a mixture component), as well as split or merge moves that partition or combine components. Other moves may rely on problem-specific reparameterizations to achieve dimension matching. Priors are specified over models and their parameters, and the likelihood is the same as in fixed-dimension MCMC.

The acceptance probability follows a form analogous to Metropolis-Hastings, but includes model priors, proposal densities across models, and the Jacobian of the dimension-changing transformation. RJMCMC can be computationally intensive and requires careful design of proposals to ensure good mixing. It is widely used for Bayesian model selection in mixture models, regression with variable selection, change-point problems, phylogenetic inference, and other settings where the model structure is uncertain.