MCMC for Model Comparison
Markov Chain Monte Carlo for Bayesian Model Comparison · Also known as: reversible-jump MCMC, RJMCMC, marginal likelihood estimation via MCMC, Bayesian model selection via MCMC
MCMC for model comparison uses Markov chain Monte Carlo algorithms to estimate the marginal likelihoods and Bayes factors needed to formally compare competing statistical models. Techniques such as reversible-jump MCMC and bridge sampling allow exploration across model spaces of different dimensionality, enabling fully Bayesian model selection and averaging.
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When to use it
Use MCMC for model comparison when you need a principled, fully Bayesian comparison of models that differ in structure or dimensionality — for example, comparing regression models with different covariate sets, choosing among mixture models with different component counts, or testing competing dynamic specifications. It is particularly valuable when the models are complex enough that information criteria (AIC, BIC) are unreliable approximations. Avoid it when the number of candidate models is extremely large (model space explosion makes full enumeration infeasible) or when computation time is the binding constraint and approximate criteria suffice.
Strengths & limitations
- Provides exact (in the limit) Bayesian model probabilities rather than asymptotic approximations.
- Reversible-jump MCMC handles models of varying dimensionality in a single unified run.
- Naturally produces Bayesian model-averaged predictions weighted by posterior model probability.
- Penalises complexity automatically through marginal likelihood, without a separate penalty term.
- Compatible with any likelihood and prior, not restricted to exponential family models.
- Estimating marginal likelihoods is notoriously difficult; naive harmonic-mean estimators are unstable and should be avoided.
- Reversible-jump proposals are hard to tune: poorly designed trans-dimensional moves lead to low acceptance rates and slow mixing.
- Computationally very demanding, especially when the number of candidate models or model parameters is large.
- Results are sensitive to prior specification on model parameters, since the marginal likelihood integrates over the prior.
Frequently asked
What is a Bayes factor and how do I interpret it?
A Bayes factor BF_12 is the ratio of marginal likelihoods: how much more likely the data are under model 1 than model 2. Jeffreys' interpretive scale is commonly used: BF > 3 is moderate evidence for model 1, BF > 10 is strong evidence, BF > 100 is decisive. Values between 1/3 and 3 suggest the data do not discriminate well between the models.
Why is the harmonic mean estimator of the marginal likelihood unreliable?
The harmonic mean estimator computes the reciprocal of the average inverse likelihood over MCMC samples. Although it is consistent in principle, it has infinite variance whenever the likelihood has heavy tails relative to the posterior, which is almost always the case. Even with very long chains the estimate can be wildly off. Bridge sampling or thermodynamic integration should be used instead.
Is RJMCMC the same as Bayesian model averaging?
They are related but distinct. RJMCMC is an algorithm that samples across a joint model space simultaneously, estimating posterior model probabilities as a byproduct. Bayesian model averaging is the downstream inference step: combining predictions from multiple models weighted by their posterior probabilities. RJMCMC is one way to obtain those weights; bridge sampling estimates within each model separately are another.
When should I prefer information criteria (AIC, BIC, WAIC) over MCMC-based Bayes factors?
Information criteria are much cheaper to compute and work well when the sample size is large and models are not too complex. WAIC is a fully Bayesian predictive criterion that is easier to compute than a Bayes factor and often adequate for predictive model comparison. Full MCMC-based Bayes factors are most valuable when models differ structurally, when the sample is small, or when you need coherent posterior model probabilities for model averaging rather than a single winning model.
How sensitive are Bayes factors to the prior on model parameters?
Highly sensitive, especially with diffuse priors. Improper priors make Bayes factors undefined. Even proper but very vague priors can heavily favour simpler models (the Lindley-Jeffreys paradox). Best practice is to use weakly-informative priors grounded in the problem and to report a sensitivity analysis showing how the Bayes factor changes with reasonable prior variations.
Sources
- Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732. DOI: 10.1093/biomet/82.4.711 ↗
- Meng, X.-L., & Wong, W. H. (1996). Simulating ratios of normalizing constants via a simple identity: A theoretical exploration. Statistica Sinica, 6(4), 831–860. link ↗
How to cite this page
ScholarGate. (2026, June 3). Markov Chain Monte Carlo for Bayesian Model Comparison. ScholarGate. https://scholargate.app/en/bayesian/mcmc-for-model-comparison
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Approximate Bayesian ComputationSimulation↔ compare
- Bayesian Model AveragingBayesian↔ compare
- Gibbs SamplingBayesian↔ compare
- Hamiltonian Monte CarloBayesian↔ compare
- MCMCBayesian↔ compare