The Ball Pit Algorithm: A Markov Chain Monte Carlo Method Based on Path Integrals

Abstract: The Ball Pit Algorithm (BPA) is a novel Markov chain Monte Carlo (MCMC) algorithm for sampling marginal posterior distributions developed from the path integral formulation of the Bayesian analysis for Markov chains. The BPA yielded comparable results to the Hamiltonian Monte Carlo as implemented by the adaptive No U-Turn Sampler (NUTS) in sampling posterior distributions for simulated data from Bernoulli and Poisson likelihoods. One major advantage of the BPA is its significantly lower computational time, which was measured to be at least 95% faster than NUTS in analyzing single parameter models. The BPA was also applied to a multi-parameter Cauchy model using real data of the height differences of cross- and self-fertilized plants. The posterior medians for the location parameter were consistent with other Bayesian sampling methods. Additionally, the posterior median for the logarithm of the scale parameter obtained from the BPA was close to the estimated posterior median calculated using the Laplace normal approximation. The computational time of the BPA implementation of the Cauchy analysis is 55% faster compared to that for NUTS. Overall, we have found that the BPA is a highly efficient alternative to the Hamiltonian Monte Carlo and other standard MCMC methods.