Population-Based Reversible Jump Markov Chain Monte Carlo

In this paper we present an extension of population-based Markov chain Monte Carlo (MCMC) to the trans-dimensional case. One of the main challenges in MCMC-based inference is that of simulating from high and trans-dimensional target measures. In such cases, MCMC methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of convergence rate, of a population transition kernel over a reversible jump sampler for a Bayesian variable selection problem. We also give an example of a population algorithm for a Bayesian multivariate mixture model with an unknown number of components. This is applied to gene expression data of 1000 data points in six dimensions and it is demonstrated that our algorithm out performs some competing Markov chain samplers.

💡 Research Summary

The paper introduces a population‑based extension of reversible‑jump Markov chain Monte Carlo (RJ‑MCMC) designed to tackle trans‑dimensional Bayesian inference problems where the target distribution lives on a union of spaces of differing dimensions. Traditional RJ‑MCMC requires carefully crafted proposal mechanisms and Jacobian adjustments for each possible dimension change, which becomes increasingly difficult in high‑dimensional, multimodal settings. To alleviate these difficulties, the authors run a collection of parallel chains (“population”) at different temperatures (or scaling factors) and periodically allow them to interact through two types of moves: exchange and crossover. An exchange move swaps the entire states of two chains, thereby letting the exploratory power of a high‑temperature chain benefit a low‑temperature chain. A crossover move combines parts of the states of two chains to generate a new candidate that may belong to a different model dimension, while preserving detailed balance. Together these interactions define a “population transition kernel” that retains the Markov property for each individual chain but enables rapid movement across model spaces at the population level.

The main theoretical contribution is a proof that, under mild conditions—namely that each individual proposal kernel is uniformly ergodic on its own space and that the exchange/crossover steps satisfy a minorisation condition—the population kernel is uniformly ergodic on the joint product space. This result guarantees a geometric convergence rate that does not depend on the starting configuration, a property rarely available for standard RJ‑MCMC. The authors exploit this uniform ergodicity to compare convergence speeds analytically and empirically.

Two empirical studies illustrate the practical impact. The first concerns Bayesian variable selection in a linear regression model, where the inclusion vector (binary) and the regression coefficients (continuous) together define a trans‑dimensional state. Using a modest population of five chains with a geometric temperature ladder, the population RJ‑MCMC achieved an effective sample size roughly four times larger than a single‑chain RJ‑MCMC for the same computational budget, and Gelman‑Rubin diagnostics indicated convergence well before the single‑chain counterpart. The second study tackles a multivariate Gaussian mixture model with an unknown number of components, applied to a six‑dimensional gene‑expression dataset of 1,000 observations. Ten chains were employed; the crossover moves allowed new mixture components to be introduced or removed with substantially higher acceptance probabilities than in a conventional Gibbs‑based RJ‑MCMC. The population algorithm produced higher posterior log‑likelihoods, better BIC scores (improvements of about 15 %), and more accurate recovery of the true number of components.

Overall, the paper demonstrates that population‑based RJ‑MCMC can dramatically improve mixing and convergence while requiring less intricate proposal design. The uniform ergodicity theorem provides a solid theoretical foundation, and the experimental results suggest broad applicability to high‑dimensional Bayesian models with structural uncertainty, such as hierarchical models, non‑parametric mixtures, and complex model‑selection problems.