Reversible Jump MCMC

Definition

Reversible jump Markov chain Monte Carlo (RJMCMC), introduced by Green (1995), extends the Metropolis-Hastings algorithm to trans-dimensional parameter spaces. It constructs a single Markov chain that jumps between models of different dimension, with stationary distribution $\pi(\theta, k) \propto \pi(k)\,L(y|\theta_k)$ — a mixture over candidate dimensions $k$ . Posterior probabilities $\hat\pi(k|y)$ are estimated as the fraction of MCMC draws at dimension $k$ ; Bayes factors are their ratios.

Key Ideas

Dimension-matching bijection: to move from dimension $d$ to $d'>d$ , draw auxiliary variables $u\sim q(u)$ of dimension $d'-d$ and construct $\theta'=g(\theta,u)$ via a differentiable bijection. The reverse move uses $u'\sim q(u')$ with $\theta=g^{-1}(\theta',u')$ . The joint $(\theta,u)$ and $(\theta',u')$ must have equal dimension.
Acceptance ratio: $\alpha = \min\!\left[1,\;\frac{\pi(\theta',k')\,q(u')}{\pi(\theta,k)\,q(u)}\left|\frac{\partial(\theta',u')}{\partial(\theta,u)}\right|\right]$ . The Jacobian corrects for the change of variables and ensures detailed balance holds across model dimensions.
Move types: standard implementations include birth (add a component, $k\to k+1$ ), death (remove a component, $k\to k-1$ ), move (perturb current parameters at fixed $k$ ), and parameter renewal (e.g., Gibbs update of nuisance $\sigma^2$ ). Move probabilities $p_b(k)$ and $p_d(k)$ must satisfy the balance condition $p_b(k)\pi(k) = p_d(k+1)\pi(k+1)$ for ergodicity.
Prior on dimension: a truncated Poisson $\pi(k)\propto\text{Pois}(\lambda)$ on $\{1,\ldots,K_{\max}\}$ or geometric prior penalises large $k$ and ensures the chain is bounded.

How It Works

Original Framework and Applications (Green 1995)

Green (1995) demonstrates the framework on three problems. The joint distribution is $p(k, \theta^{(k)}, y) = p(k)p(\theta^{(k)}\mid k)p(y\mid k,\theta^{(k)})$ ; the MCMC target is the joint posterior $p(k, \theta^{(k)}\mid y)$ .

Multiple change-point (Poisson process): Coal mining disasters 1851–1962 (n=192, L=40,907 days). Prior: $k \sim \text{Poisson}(\lambda=3)$ truncated at $k_{\max}=30$ ; step positions are even-order statistics from $2k+1$ uniform draws (spacing prior); heights $h_j \sim \text{Gamma}(1,200)$ . Four move types (H, P, birth, death); birth splits height $h_j$ using $u \sim \text{Uniform}[0,1]$ via $h_{j+1}'/h_j' = (1-u)/u$ with Jacobian $(h_j'+h_{j+1}')^2/h_j$ . Result: posterior mode k=3 change-points; main break near 25 June 1890; 40,000 iterations/45 sec (Sun Sparc 2).

Image segmentation (Voronoi tessellation): 2D analogue — k tiles, each with uniform intensity; birth/death of generating points. Synthetic $50\times50$ disc (intensity 2.0 vs. background 0.5, $\sigma=0.7$ ) recovered after 20,000 sweeps, 260 sec.

Partition models (binomial experiments): Pine seedling mortality data (Consonni-Veronese 1995), 4 responses in a $2\times 2$ factorial; groups share a common $\beta$ drawn from a prior partition; RJMCMC results replicate Consonni-Veronese analytic approximations.

Triangular Mixture Application (Perron-Mengersen 2001)

Model: nondecreasing regression $X_i=F(t_i)+\varepsilon_i$ , $\varepsilon_i\sim\mathcal{N}(0,\sigma^2)$ , with $F$ approximated as a mixture of $r$ triangular cumulative distribution functions (CDFs) on partition $\{x_1,\ldots,x_{r-1}\}\subset(0,1)$ . Parameters $\theta=(\sigma^2, r, x_1,\ldots,x_{r-1})$ . Four operations are proposed at each step:

Renew $\sigma^2$ : Gibbs draw from $\pi(\sigma^2|r,x)$ .
Death: draw node index $j$ uniformly; propose $\underline\theta_0=(\sigma^2,r-1,\text{partition without }x_j)$ ; accept with probability $\min[1,\;\lambda(x_{j+1}-x_{j-1})\,p_b(r-1)\,f(y|\underline\theta_0) / (p_d(r)\,f(y|\theta))]$ .
Move: draw $j$ uniformly; propose $z\sim\text{Uniform}[x_{j-1},x_{j+1}]$ ; standard Metropolis-Hastings (MH) accept ratio $f(y|\theta_z)/f(y|\theta)$ .
Birth: draw $j$ uniformly; insert $z\sim\text{Uniform}[x_{j-1},x_j]$ ; accept with probability $\min[1,\;p_d(r+1)\,f(y|\underline\theta_0) / (\lambda(x_j-x_{j-1})\,p_b(r)\,f(y|\theta))]$ .

In simulations the posterior for $r$ concentrates between 10 and 30 with unimodal, roughly symmetric shape.

Bayes Factor Computation (Han-Carlin 2001)

For an acquired immunodeficiency syndrome (AIDS) CD4 count model (467 subjects), Carlin-Chib product-space and modified Carlin-Chib methods fail to converge after $1.5\times10^6$ iterations. RJMCMC achieves $\hat B_{21}=67.83$ — strong evidence for the random-effects model — and is the recommended approach for variable-dimension problems.

Why It Matters

Marginalises over model dimension within a single MCMC run rather than requiring separate runs per candidate $k$ or a product-space augmentation (Carlin-Chib 1995).
Well-suited to nonparametric problems where the effective dimension is unknown: number of spline knots, mixture components, change-point locations, or regression terms.
Posterior $\hat\pi(k|y)$ directly estimates model probabilities; no normalising constant is needed for Bayes factors between models of the same type.

Open Questions

Choosing the bijection $g$ and proposal $q(u)$ to achieve efficient mixing is problem-specific and requires tuning; poor choices result in near-zero acceptance for birth/death moves.
Convergence diagnostics for trans-dimensional chains are harder than for fixed-dimension MCMC since standard spectral methods apply only within each $k$ .
For small numbers of candidate models, running Chib (1995) or Chib-Jeliazkov (2001) for each $k$ separately may be more efficient and easier to diagnose.