Reversible Jump MCMC

• RJMCMC is a Bayesian inference method that handles trans-dimensional model selection by sampling joint posteriors over models and parameters, preserving detailed balance.
• It employs both within-model proposals and trans-dimensional moves using auxiliary variables and Jacobian determinants to achieve accurate dimension matching.
• Advanced implementations incorporate birth/death, split/merge, and transport moves with adaptive tuning to enhance acceptance rates and computational efficiency.

Reversible Jump Markov Chain Monte Carlo (RJMCMC) is a Metropolis–Hastings–based Markov chain Monte Carlo framework designed for Bayesian inference in problems where the number of parameters—i.e., the dimension of the parameter space—is itself an unknown to be estimated, such as in model selection, mixture modeling, or variable selection. RJMCMC enables rigorous sampling from the joint posterior distribution over model indicators and model-specific parameter vectors, preserving detailed balance even as the chain moves across parameter spaces of differing dimensions. This approach generalizes fixed-dimensional MCMC to trans-dimensional and even "trans-structural" inference, and serves as the backbone of state-of-the-art Bayesian multimodel inference procedures.

1. Foundations and Acceptance Mechanism

RJMCMC constructs a Markov chain on a state space defined as the union of all model-parameter spaces $$X = \bigcup_{k \in \mathcal{K}}\lbrace k \rbrace \times Z_k$$, where $k$ indexes the model and $Z_k \subset \mathbb{R}^{d_k}$ is the model’s parameter space. The sampler targets a posterior $$\pi(k, z_k) \propto \pi(k) \, \pi(z_k \mid k) \, L(y \mid z_k, k)$$, and alternates between:

• Within-model moves: parameter proposals at fixed $k$ (standard MH or Gibbs steps)
• Between-model (trans-dimensional) moves: proposals to a new model $k'$, with appropriate construction of a parameter mapping to ensure dimension-matching

The acceptance probability for a trans-dimensional move from $(k, z)$ to $(k', z')$ is given by

$$\alpha\bigl( (k, z) \to (k', z') \bigr) = \min\left\{1, \frac{ \pi(k', z' \mid y) \, q(k' \to k) \, q_{k'\to k}(u') }{ \pi(k, z \mid y) \, q(k \to k') \, q_{k \to k'}(u) } \; \bigg| \det \frac{\partial (z', u')}{\partial (z, u)} \bigg| \right\}$$

where $u, u'$ are auxiliary random variables introduced to match the total number of dimensions, $q_{k \to k'}$ is the proposal kernel for auxiliary variables, and the determinant is the Jacobian of the dimension-matching transformation (Fan et al., 2010).

2. Dimension Matching, Move Types, and Jacobians

A core theoretical requirement of RJMCMC is the dimension-matching condition: for a move from $k$ to $k'$, auxiliary variables $u$ (dimension $d_{k \to k'}$) and $u'$ (dimension $d_{k' \to k}$) are constructed such that $d_k + d_{k \to k'} = d_{k'} + d_{k' \to k}$ (Fan et al., 2010, Karakuş et al., 2017). Common implementations include:

• Birth/Death moves: add or drop a parameter (e.g., adding a mixture component), typically with $u$ being the "new" parameter; for simple concatenation, the Jacobian is unity.
• Split/Merge and more general moves: e.g., splitting a mixture component, where $u$ encodes randomization in the split, and the Jacobian must be computed explicitly.

For example, in geophysical inversion, birth moves are realized by introducing new polygon vertices with carefully constructed proposals and explicit calculation of the Jacobian to ensure correct invariant distribution (Luo, 2010).

3. Proposals, Algorithmic Structure, and Practical Tuning

In practice, the design of within-model and between-model proposals is crucial for computational efficiency:

• Within-model proposals: often standard random-walk or adaptive proposals, potentially leveraging local curvature (e.g., via negative Hessian at MLE) (Bouranis et al., 2017).
• Between-model proposals: parameter proposals may use independent draws (e.g., from fitted Gaussian surrogates, or "palette" mappings (Barker et al., 2010)) or more sophisticated strategies such as local surrogate fitting (Gagnon, 2019), interpolation from single-model MCMC samples (Farr et al., 2011), or proposals based on variational normalizing flows (Yin et al., 14 Dec 2025).
• Automated and adaptive tuning: several schemes adapt proposal covariances (Tian et al., 13 Feb 2024), proposal weights, and temperatures (for tempering) to maintain targeted acceptance rates (e.g., 0.234-rule for optimal scaling (Gagnon et al., 2016)).

An archetypal RJMCMC iteration consists of selecting a move type (within- or between-model, possibly split/merge), sampling any required auxiliary variables, mapping parameters to the new space, evaluating the acceptance ratio (including Jacobian), and accepting/rejecting the new state per Metropolis–Hastings logic (Fan et al., 2010, Bouranis et al., 2017, Stroeer et al., 2009).

4. Extensions and Algorithmic Innovations

The generic RJMCMC scheme has motivated substantial methodological enhancements:

• Noisy RJMCMC for doubly-intractable posteriors: When model likelihoods are unavailable analytically (e.g., Gibbs random fields), unbiased estimators of intractable quantities are introduced. The algorithm substitutes exact ratios with MC-estimated ratios (e.g., via importance sampling or telescopic product estimators), with controlled variance and convergence results depending on estimator variance (Bouranis et al., 2017).
• Multiple-Try RJMCMC (GMTRJ): At each model jump, several candidate proposals are generated, one is selected via weighted probabilities informed by local Gaussian approximations, and detailed balance is preserved via adjusted acceptance ratios. The result is increased acceptance rates and reduced autocorrelation, especially in high-dimension or poorly scaled settings (Pandolfi et al., 2010).
• Trans-space RJMCMC: Moves are allowed not just between models of different dimension, but also between families of structurally different models (e.g., moving between Gaussian, α-stable, and t-distributions), with invertible mappings bridging parameter spaces and explicit Jacobians for cross-family proposals (Karakuş et al., 2017).
• Transport RJMCMC via Normalizing Flows: Model-specific normalizing flows are optimized (reverse KL) to approximate conditional posteriors, enabling highly informed within- and between-model proposals. Such approaches lead to improved marginal likelihood estimation and superior mixing in complex model spaces (Yin et al., 14 Dec 2025).
• Sequential Monte Carlo with Transformations (TSMC): RJMCMC's accept/reject paradigm is replaced by dimension-matching transformations within SMC, leveraging population weighting instead of rejection, leading to low-variance evidence estimates and better scalability for complex model-comparison problems (Everitt et al., 2016).

5. Convergence Theory and Ergodicity

The theoretical ergodicity properties of RJMCMC are established through a combination of Markov chain decomposition and drift/minorization arguments:

• If within-model kernels (e.g., Metropolis–Hastings, Gibbs) are geometrically ergodic for each fixed $k$ and the between-model proposal kernel is irreducible on the model index set, the overall RJMCMC algorithm is $L^2(\pi)$-geometrically ergodic (Qin, 2023).
• The minimal conditions required are explicit: each P_k must contract functions orthogonal to the within-model invariant measure, proposals across models must allow the chain to access the full model graph, and mixture weights must remain bounded away from zero (Qin, 2023).
• Diffusion-limit arguments establish optimal scaling and move probabilities: the 0.234-rule for within-model proposal scaling is recovered in the RJMCMC context, and the optimal probability of proposing a parameter update versus a trans-dimensional move is $O(1/\sqrt{A})$ where $A$ controls the maximal discrepancy between proposal and target densities (Gagnon et al., 2016).

6. Applications and Efficiency Considerations

RJMCMC has found broad application domains including mixture modeling, variable selection in regression, nonparametric function estimation, gravitational wave signal analysis, geophysical inverse problems, nonparametric drift estimation for diffusions, light transport in computer graphics, and analysis of stochastic movement models (Stroeer et al., 2009, Luo, 2010, Luo et al., 2013, Meulen et al., 2012, Blackwell, 25 Sep 2025, Bitterli et al., 2017).

Performance considerations are dominated by the construction of efficient, well-matched proposal mechanisms and accurate calculation of the acceptance ratio (including Jacobians). Algorithmic efficiency is frequently improved via adaptive schemes, parallelization (e.g., for telescopic estimators (Bouranis et al., 2017)), and use of informed or surrogate proposal distributions (Gagnon, 2019, Yin et al., 14 Dec 2025). Data-driven adaptive transport maps, and post-hoc interpolation schemes also dramatically increase mixing and effective sample size, especially as model dimension or structural heterogeneity increases (Farr et al., 2011, Yin et al., 14 Dec 2025).

7. Practical Guidelines and Empirical Observations

Key practical insights for implementing RJMCMC include:

• Proposal distributions for new parameters should be as close to the local posterior as possible (e.g., using Laplace, empirical posterior, or normalizing flow approximations) (Gagnon, 2019, Yin et al., 14 Dec 2025).
• Between-model proposal probabilities should be adapted to balance exploration and acceptance, with adaptive weights and mechanisms informed by pilot MCMC or proxy marginal likelihood estimates (Stroeer et al., 2009, Tian et al., 13 Feb 2024).
• Careful tuning of step sizes (for within-model moves) and move-type probabilities (for switching) improves overall efficiency; empirical results confirm the theoretical efficiency gains of optimally tuned samplers (Gagnon et al., 2016).
• Comprehensive convergence diagnostics—including effective sample size, mixing rates of the model indicator, and variance of model-probability estimates—are essential. Empirically, advanced proposals (e.g., informed, multiple-try, transport) routinely yield increases in acceptance rates and reductions in estimator variance by factors of 2–10 or more (Pandolfi et al., 2010, Yin et al., 14 Dec 2025, Tian et al., 13 Feb 2024).

RJMCMC remains the reference algorithm for rigorous Bayesian model selection and evidence estimation in trans-dimensional and multimodel contexts. Its current state of the art combines the foundational Metropolis–Hastings machinery with sophisticated proposal adaptation, surrogate modeling, and hybrid MCMC/SMC methodologies (Fan et al., 2010, Everitt et al., 2016, Bouranis et al., 2017, Gagnon, 2019, Yin et al., 14 Dec 2025).