Assessing the Application of Markov Chain Monte Carlo Methods in Bayesian Statistical Computation and Inference

Posted: Nov 20, 2017

• Daniel Young
• David Allen
• David Garcia

Abstract

Bayesian statistical methods have experienced a remarkable resurgence in recent decades, largely driven by advances in computational techniques that enable practical implementation of complex models. At the heart of this computational revolution lie Markov Chain Monte Carlo (MCMC) methods, which provide a powerful framework for approximating posterior distributions that are analytically intractable. The fundamental principle underlying MCMC involves constructing a Markov chain that has the desired posterior distribution as its stationary distribution, thereby allowing researchers to generate samples from complex probability distributions through iterative sampling algorithms. Despite their widespread adoption and theoretical elegance, conventional MCMC methods face significant challenges in practical applications. The curse of dimensionality presents a particularly formidable obstacle, as the efficiency of random walk-based samplers deteriorates rapidly with increasing parameter dimensions. Additionally, multimodal posterior distributions often trap chains in local modes, leading to incomplete exploration of the parameter space and biased inference. These limitations become especially pronounced in modern applications involving high-dimensional data, complex hierarchical models, and applications requiring real-time inference. This research addresses these challenges through the development and evaluation of a novel hybrid MCMC framework that integrates Hamiltonian dynamics with adaptive temperature control. Our approach builds upon the theoretical foundations of existing methods while introducing innovative modifications that enhance both computational efficiency and statistical reliability. By dynamically adjusting the temperature parameter during sampling, our method