A Detailed Implementation of Reversible Jump Markov Chain Monte Carlo (RJMCMC) Sampling Algorithm

In this article, we present a detailed implementation of a Reversible Jump Markov Chain Monte Carlo (RJMCMC) sampling algorithm, accompanied by relevant research papers to facilitate comparative study and deeper algorithmic understanding. The implementation incorporates multiple proposal mechanisms including Birth, Death, Split, Merge, and Update operations. The Birth mechanism introduces new particles into the system through specific proposal distributions, while the Death mechanism removes existing particles with carefully designed reverse transitions. The Split operation divides a single particle into two new entities with dimension-matching transformations, and the Merge operation combines two particles into one with corresponding Jacobian adjustments for dimensional changes. The Update mechanism modifies particle states while maintaining dimensional consistency. Through strategic combination of these transition operators, the algorithm achieves efficient sampling across variable-dimensional spaces, with widespread applications in statistical physics, chemical modeling, and biological systems. The code implementation includes detailed balancing checks and acceptance probability calculations using Green's formalism for dimension-changing moves. We further analyze the algorithm's strengths in handling model uncertainty and limitations in high-dimensional spaces, while discussing potential enhancements like adaptive proposal distributions and parallel tempering techniques for broader application scenarios.