Reversible-jump Markov chain Monte Carlo

Reversible-jump Markov chain Monte Carlo samples a posterior on a disjoint union of model-specific parameter spaces. A dimension-matching bijection between old parameters plus auxiliary variables and new parameters plus reverse auxiliary variables supplies a reversible proposal. Its acceptance ratio includes posterior-density, model-selection, proposal-density, and Jacobian determinant factors. The equal-dimension reversible-jump acceptance probability is the ordinary density-based special case.

A pseudo-prior is a proper density assigned to a parameter that is inactive under a particular model. Integrating it out leaves that model's likelihood and marginal evidence unchanged. Its choice affects movement between model states, but not the intended marginal posterior over models.

A pseudo-prior can make model parameter vectors have equal dimensions, allowing identity-matched RJ-MCMC moves. If active and inactive coordinates share the same prior density in the two augmented targets, these factors cancel in model-switch ratios. Within-model updates must preserve the augmented target, including the inactive coordinate's pseudo-prior law.

Compare independent Poisson means with one shared mean under proper shape-rate gamma distribution priors. Adding the second mean as an inactive parameter under the shared-mean model gives a simple identity model switch. With equal model priors and identical active/inactive priors, its ratio from separate to shared means is $e^{{\gamma }_2-{\gamma }_1}({\gamma }_1/{\gamma }_2{)}^{x_2}$. Poisson-gamma conjugacy supplies exact within-model updates, and integrating the gamma kernels supplies an independent Bayes factor benchmark.

For unnormalized model posterior densities $h_j$, model-selection probabilities $s_{jk}$ and parameter-proposal densities $q_{jk}$, accept a proposal $(j,\theta )\to (k,{\theta }^{\prime })$ with the minimum of one and $h_k({\theta }^{\prime })s_{kj}({\theta }^{\prime })q_{kj}(\theta \mid {\theta }^{\prime })/[h_j(\theta )s_{jk}(\theta )q_{jk}({\theta }^{\prime }\mid \theta )]$. A deterministic matching-map formulation instead includes its explicit Jacobian determinant; it must not be counted again when already absorbed into a direct proposal density.