RJMCMC Algorithm with Resampling Step: Enhanced Bayesian Model Selection

The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm is an extended MCMC method for Bayesian model selection and parameter estimation. Building upon traditional MCMC, it introduces cross-model space jumping capability through reversible transition mechanisms that enable sampling across parameter spaces of different dimensions.

Core Mechanisms Cross-model transitions: Design transition kernels satisfying detailed balance conditions, allowing chains to jump between models of varying complexity. In implementation, this typically involves creating proposal distributions that maintain reversibility through dimension-matching functions. Dimension matching: When proposing jumps from simpler to more complex models, random variables are used to supplement dimension differences. Code implementation often requires generating auxiliary variables to match parameter space dimensions between models. Acceptance probability calculation: Includes Jacobian determinant terms to compensate for metric differences between parameter spaces. The acceptance ratio calculation must account for the probability of reverse moves to maintain detailed balance.

Resampling Enhancement Traditional RJMCMC suffers from inefficient model space exploration. The enhanced version with resampling: Periodically preserves high-weight samples and reinitializes inefficient chain segments through systematic resampling procedures Maintains sample diversity using residual resampling mechanisms Adaptively adjusts inter-model transition probabilities based on recent acceptance rates Implementation typically involves monitoring chain efficiency metrics and triggering resampling when convergence slows.

Performance Validation Methods Convergence diagnostics: Observe cross-model acceptance rates stabilizing within the 15-30% optimal range Mixing speed: Compare iteration counts required to reach target distributions using Gelman-Rubin statistics Model probability estimation: Examine variance decay rates of posterior model probabilities through trace plot analysis

Typical testing scenarios employ Gaussian mixture model selection, using simulated data to verify algorithm's ability to identify true component numbers. The enhanced resampling version should demonstrate: fewer mode omissions, faster model space exploration, and more stable sampling performance in multimodal distributions. Code validation often involves comparing marginal likelihood estimates across different resampling strategies.