Polya tree Monte Carlo method

Abstract

Markov Chain Monte Carlo (MCMC) methods have been widely used in Statistics and machine learning research. However, such methods have several limitations, including slow convergence and the inefficiency in handling multi-modal distributions. To overcome these limitations of MCMC methods, a new, efficient sampling method has been proposed and it applies to general distributions including multi-modal ones or those having complex structure. The proposed approach, called the Polya tree Monte Carlo (PTMC) method, roots in constructing a Polya tree distribution using the idea of Monte Carlo method, and then using this distribution to approximate and facilitate sampling from a target distribution that may be complex or have multiple modes. The associated convergence property of the PTMC method is established and computationally efficient sampling algorithms are developed based on the PTMC. Extensive numerical studies demonstrate the satisfactory performance of the proposed method under various settings including its superiority to the usual MCMC algorithms. The evaluation and comparison are carried out in terms of sampling efficiency, computational speed and the capacity of identifying distribution modes. Additional details about the method, proofs and simulation results are provided in the Supplementary Web Appendices online.