Green (1995) Reversible Jump MCMC and Bayesian Model Determination

Summary

Green (1995) proposes a general framework — reversible jump Markov chain Monte Carlo (RJMCMC) — for constructing reversible Markov chain samplers that jump between parameter subspaces of differing dimensionality. The central innovation is dimension matching: when proposing a move from a model of dimension $n_1$ to one of dimension $n_2$, auxiliary random variables $u$ are drawn so that $n_1+m_1 = n_2+m_2$, enabling a differentiable bijection and ensuring detailed balance via a Jacobian correction. Three applications are demonstrated: multiple change-point analysis for a Poisson process (coal mining disasters 1851–1962), two-dimensional image segmentation via Voronoi tessellation, and Bayesian partition models for binomial experiments.

Key Claims

• Standard MCMC is restricted to fixed-dimensional parameter spaces; RJMCMC extends the Metropolis-Hastings framework to trans-dimensional model spaces $\{M_k, k \in K\}$ where the joint distribution is $p(k, \theta^{(k)}, y) = p(k)\,p(\theta^{(k)}|k)\,p(y|k, \theta^{(k)})$.
• Dimension-matching bijection: to jump from dimension $n_1$ to $n_2$, draw auxiliary variables $u^{(1)}$ (length $m_1$) and $u^{(2)}$ (length $m_2$) with $n_1+m_1 = n_2+m_2$, making $(\theta^{(1)}, u^{(1)}) \leftrightarrow (\theta^{(2)}, u^{(2)})$ a bijection $g$.
• Acceptance probability for a trans-dimensional move $x=(1,\theta^{(1)}) \to x'=(2,\theta^{(2)})$ (eq. 7): $$\alpha = \min\left\{1,\; \frac{p(2,\theta^{(2)}|y)\,j(2,\theta^{(2)})\,q_2(u^{(2)})}{p(1,\theta^{(1)}|y)\,j(1,\theta^{(1)})\,q_1(u^{(1)})}\left|\frac{\partial(\theta^{(2)},u^{(2)})}{\partial(\theta^{(1)},u^{(1)})}\right|\right\}$$
• Jump-diffusion (Grenander-Miller 1994) only approximately maintains detailed balance in within-subspace moves; Carlin-Chib (1995) product-space approach requires continual updating of irrelevant parameters, limiting scalability; embedding methods are cumbersome.
• Prior distributions across subspaces need not be individually normalised, but relative normalising constants must be consistent — a constraint Grenander & Miller (1994) overlooked.
• The sampler is entirely constructive: no integration is needed to set up the transition mechanism (Remark 1, p. 715).
• Multiple move types are combined at random to form a hybrid sampler (Tierney 1994); detailed balance is enforced within each move type separately.
• Change-point application (§4): piecewise-constant Poisson rate on $[0,L]$; $k \sim \text{Poisson}(\lambda)$, truncated at $k_{\max}$; step positions are even-order statistics from $2k+1$ uniform $[0,L]$ draws (spacing prior); heights $h_j \sim \text{Gamma}(\alpha,\beta)$. Move types: H (height), P (position), birth ($k \to k+1$), death ($k \to k-1$). Birth splits height $h_j$ of the interval containing $s^*$ into $h_j', h_{j+1}'$ via $u \sim \text{Uniform}[0,1]$ with $h_{j+1}'/h_j' = (1-u)/u$; the Jacobian is $(h_j'+h_{j+1}')^2/h_j$.
• Coal mining results: $n=192$ disasters, $L=40{,}907$ days, $\lambda=3$, $\alpha=1$, $\beta=200$, $k_{\max}=30$; 40,000 iterations after 4,000 burn-in (45 sec, Sun Sparc 2); posterior mode $k=3$; main change-point near day 14420 ≈ 25 June 1890 (95% credible interval: 24 May 1887–7 May 1896).
• Image segmentation (§5): Voronoi tessellation with k tiles; birth/death of generating points; synthetic $50\times50$ disc image recovered after 20,000 sweeps, 260 sec.
• Partition models (§6): pine seedling mortality data ($n=4$ binomial responses, $2\times2$ factorial; Consonni-Veronese 1995); RJMCMC results replicate analytic approximations; random-$q$ extension posterior mean $\hat q=181$, standard deviation (SD) $=58$.

Concepts Introduced or Extended

• Reversible Jump MCMC
• Change-Point Analysis

Entities Mentioned

• Peter Green

Quotes

"This paper proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of differing dimensionality, which is flexible and entirely constructive." (p. 711)

"The definition of the sampling method is entirely constructive. No integration, by simulation or otherwise, is needed to set up the transition mechanism." (Remark 1, p. 715)

"Although as usual with Hastings methods, the distribution π need not be normalised, relative normalising constants between different subspaces are needed … a point apparently missed by Grenander & Miller (1994)." (Remark 3, p. 715)

My Take

This is the foundational paper for RJMCMC — cited in virtually every Bayesian variable-dimension paper since 1995. The key insight is elegant: add auxiliary variables to equate dimensions, then use a standard Jacobian acceptance ratio. The main practical challenge — choosing the bijection g and proposal q(u) for efficient mixing — is left problem-specific, and Green acknowledges no claim of optimality for his particular choices. The coal mining example has become a standard benchmark. The paper's three-application breadth (1D change-point, 2D image segmentation, partition models) is effective. The convergence diagnostics problem for trans-dimensional chains, acknowledged by Green in §7, remains an active area of research. Richardson-Green (1997) later applied RJMCMC to Bayesian mixture estimation with unknown k — the canonical follow-up.