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.

To minimize $E_j(\theta )=2k_j-2{\ell }_j(\theta )$ over models with different parameter dimensions, sample from a target proportional to $w_j{e}^{-E_j(\theta )/(2T)}$ and cool $T$. A dimension-matching proposal augments the smaller model by auxiliary variables, transforms them bijectively into the larger parameter vector, and includes the proposal probability density function and absolute Jacobian determinant in the Metropolis–Hastings acceptance probability. For a positive Poisson mean $\lambda$ and normal parameters $(\mu ,v)$, one convenient transformation isIts absolute Jacobian determinant is one. If the forward and reverse model-jump selection probabilities are $b$ and $d$, the Poisson-to-normal ratio isThe reverse ratio is its reciprocal at the inverse map. This enforces detailed balance. When the target is proper and the annealing chain explores it adequately, the low-temperature laws concentrate on the minimum Akaike information criterion, rather than the maximum marginal model evidence. Likelihoods compared across models must describe the same observed data with a compatible reference measure.

For one-parameter binomial and two-parameter normal models, the displayed dimension-matching map has inverse $(p,u)=((1+e^{-\mu }{)}^{-1},\log v)$ and absolute Jacobian determinant $v/[p(1-p)]$. With auxiliary density $h(u)$ and model-jump probabilities $b,d$, an annealing target proportional to $e^{({\ell }_j-k_j)/T}$ gives forward acceptance ratioThe reverse ratio is its reciprocal at the inverse map. Keep model-dependent likelihood constants. Comparing continuous densities directly with discrete masses is not invariant to measurement units; a scientifically meaningful comparison needs models for the same recorded observations with a compatible reference measure, for example bin-integrated normal probabilities for integer-valued measurements.

For nested regression models, append one coefficient $u$ drawn from density $g_k$ while retaining the variance $v$ and existing coefficients $a$. The dimension-matching map $(a,v,u)\mapsto (a,u,v)$ has absolute Jacobian one. With joint model/parameter target $t_k$, birth-selection probability $b_k$, and reverse death-selection probability $d_{k+1}$, the displayed acceptance probability and its reciprocal death rule enforce detailed balance. Include normalized parameter-prior densities and a model-order prior in $t_k$: constants that depend on model order cannot be dropped from a cross-model ratio.

A polynomial regression birth move can introduce a coefficient $z$ while shifting the intercept by $-cz$, with $c$ the sample average of the new monomial. The fitted-value increment is $z(x^{k+1}-c)$, whose sample average vanishes. The Jacobian determinant of the coefficient transformation is one. A reversible-jump Markov chain Monte Carlo acceptance ratio must still include the normalized dimension-dependent prior distributions, the model-order prior distribution, the birth/death selection probabilities and the proposal density for $z$.

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.