What is Reversible Jump MCMC?
Reversible Jump MCMC is a trans-dimensional Markov Chain Monte Carlo method that samples a model indicator and its parameters by constructing reversible moves between spaces of different dimension.
Quick Facts
| Specification | Official Specification |
|---|
How It Works
Sample over a union of parameter spaces
Write the state as (k, theta_k), where model k has parameter space Theta_k. The target is a joint posterior over union_k {k} x Theta_k, not one fixed Euclidean vector. Within-model moves can use ordinary MCMC, while between-model moves change both k and the representation of theta_k.
Green's original RJMCMC paper introduced a constructive reversible framework for Bayesian model determination. Posterior model occupancy is meaningful only under the declared model and parameter priors.
Match dimensions and include the Jacobian
A move from model k to k' augments the current parameters with random variables u, then applies a bijection (theta_k, u) -> (theta_k', u') whose input and output dimensions match. The acceptance ratio combines the target ratio, forward and reverse move probabilities, auxiliary densities, and the absolute determinant of the transformation Jacobian.
When the transformation is identity and auxiliary densities cancel, the ratio may look simple. That special case does not justify dropping the Jacobian or reverse proposal in a general split, merge, birth, or death move.
Validate movement within and between models
A chain can mix well inside every model yet almost never cross model boundaries. Record attempts and acceptances by move type, model-index traces, dwell times, occupancy, within-model ESS, parameter summaries conditional on model, and sensitivity to model priors and proposal maps.
Label switching, weakly identified components, and poorly matched birth/death proposals can distort practical exploration. Compare simulated cases with known model probabilities and, where tractable, marginal-likelihood or fixed-dimensional alternatives rather than treating visit counts as self-validating evidence.
Key Characteristics
- Samples jointly over model identity and model-specific parameters
- Supports parameter spaces whose dimension changes between states
- Uses bijective transformations with matched auxiliary dimensions
- Includes a Jacobian determinant in trans-dimensional acceptance ratios
- Combines within-model and between-model transition kernels
- Depends strongly on model priors and reversible proposal design
Common Use Cases
- Bayesian variable selection with an unknown active set
- Mixture models with an unknown number of components
- Change-point models with an unknown number of segments
- Tree and graph structures with variable complexity
- Scientific inverse problems with trans-dimensional parameterizations
Example
Loading code...Frequently Asked Questions
How does RJMCMC differ from ordinary Metropolis-Hastings?
Ordinary Metropolis-Hastings usually moves within one fixed-dimensional space. RJMCMC extends the acceptance rule to a union of model spaces and adds dimension matching, auxiliary-variable densities, move-selection probabilities, and a transformation Jacobian.
Why is dimension matching required in Reversible Jump MCMC?
A reversible proposal needs a bijection between the current parameters plus auxiliary randomness and the proposed parameters plus reverse auxiliary variables. Equal total dimensions make the change-of-variables density and its Jacobian well defined.
Can RJMCMC visit frequencies be read as posterior model probabilities?
Only after establishing valid moves, adequate cross-model mixing, and defensible model and parameter priors. Poor proposals or isolated model regions can make finite visit frequencies misleading even when the acceptance formula is correct.
Does RJMCMC solve label switching in mixture models?
No. Components with exchangeable labels can create symmetric modes within each model dimension. Identifiability constraints, label-invariant summaries, or post-processing may still be required in addition to trans-dimensional moves.
How should between-model moves be diagnosed?
Report attempts and acceptances per move type, model-index traces, dwell times, transitions between important models, conditional parameter ESS, multiple-chain agreement, and sensitivity to model priors and proposal transformations.