New Diagnostic Assessment of MCMC Algorithm Effectiveness, Efficiency, Reliability, and Controllability

Abstract

Markov Chain Monte Carlo (MCMC) is a robust statistical approach for estimating posterior distributions. However, the significant computational cost associated with MCMC presents a considerable challenge, complicating the selection of an appropriate algorithm tailored to the specific problem at hand. This study introduces a novel and comprehensive framework for evaluating the performance of MCMC algorithms, drawing inspiration from diagnostics used for multi-objective evolutionary algorithms. We employ visualizations to evaluate key algorithmic characteristics: Effectiveness (the ability to accurately find representative posterior modes, quantified by the Kullback-Leibler Divergence (KLD) and Wasserstein Distance (WD)), Efficiency (the speed of posterior characterization), Reliability (consistency across different random seeds), and Controllability (insensitivity to hyperparameter variation). Evaluating three prominent MCMC algorithms—Metropolis-Hastings (MH), Adaptive Metropolis (AM), and Differential Evolution Adaptive Metropolis (DREAM)—on high-dimensional and bimodal test problems, our analysis uncovers several insights. First, across algorithms, the number of function evaluations most controls performance on the high-dimensional problem, while the number of chains most controls performance on the bimodal problem. While this suggests similar controllability across algorithms, differences emerge on the other algorithmic characteristics. For high numbers of functions evaluations, AM performs best on the high-dimensional problem, while for low (< 5) and high (>15) chain counts, MH and AM perform best on the bimodal problem, as measured by KLD. However, outside these specific cases, DREAM consistently demonstrates superior efficiency and reliability, making it a robust choice for both high-dimensional and multimodal problems. These findings can inform MCMC algorithm selection for Bayesian inference applications, as well as hyperparameterization of the chosen algorithm. More importantly, the diagnostics represent a generalizable contribution to research on MCMC diagnostics that can be used to evaluate and inform the design of new algorithms.

Keywords

• Bayesian estimation
• Markov chain Monte Carlo
• high dimensionality
• multi-modality
• model diagnostics