[Paper Review] Asymptotically Stable Drift and Minorization for Markov Chains with Application to Albert and Chib's Algorithm

Qian Qin, James P. Hobert|arXiv (Cornell University)|Dec 24, 2017

Bayesian Methods and Mixture Models30 references3 citations

TL;DR

This paper develops refined drift and minorization conditions for Markov chain Monte Carlo (MCMC) convergence analysis in high-dimensional Bayesian problems, focusing on Albert and Chib's algorithm for the probit model. By centering the drift function and suppressing high-dimensionality in minorization, it establishes geometric ergodicity with convergence rate bounded away from 1 as both sample size $n \to \infty$ and number of covariates $p \to \infty$, yielding the first computable bounds on total variation distance to stationarity.

ABSTRACT

The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, $n$, and/or number of covariates, $p$. Recent work suggests that, while often useful for establishing convergence rates when $n$ and $p$ are fixed, techniques based on drift and minorization may not be versatile enough to handle the more delicate task of convergence complexity analysis. This article provides some general ideas for sharpening drift and minorization conditions, particularly in cases where~$n$ and/or~$p$ are large. The key ideas include developing an appropriately centered drift function, and suppressing high-dimensionality in the construction of minorization conditions. These concepts are employed in a thorough convergence complexity analysis of Albert and Chib's (1993) data augmentation algorithm for the Bayesian probit model. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as $n ightarrow \infty$ (with $p$ fixed), and as $p ightarrow \infty$ (with $n$ fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

Motivation & Objective

• To address limitations of traditional drift and minorization techniques in convergence complexity analysis for high-dimensional MCMC algorithms.
• To develop sharper, scalable drift and minorization conditions that remain effective as sample size $n$ and number of covariates $p$ increase.
• To apply these refined conditions to Albert and Chib's (1993) data augmentation algorithm for Bayesian probit regression.
• To derive computable bounds on the total variation distance to stationarity for this algorithm.
• To establish geometric ergodicity with convergence rate uniformly bounded away from 1 in both $n \to \infty$ and $p \to \infty$ regimes.

Proposed method

• Introduces a centered drift function to improve stability and scalability in high-dimensional settings.
• Modifies minorization conditions by suppressing dependence on high-dimensional components, enhancing tractability.
• Applies the refined drift and minorization framework to the specific structure of Albert and Chib's data augmentation algorithm.
• Uses the drift and minorization conditions to establish geometric ergodicity of the Markov chain.
• Derives explicit, computable bounds on the total variation distance to stationarity using the asymptotic analysis.
• Employs asymptotic analysis to verify convergence rates remain bounded away from 1 as $n \to \infty$ and $p \to \infty$.

Experimental results

Research questions

• RQ1 Can drift and minorization conditions be refined to handle convergence complexity analysis in high-dimensional Bayesian models?
• RQ2 Does Albert and Chib's data augmentation algorithm for the probit model maintain geometric ergodicity as both sample size $n$ and number of covariates $p$ grow?
• RQ3 Can computable bounds on the total variation distance to stationarity be derived using asymptotic drift and minorization analysis?
• RQ4 How does centering the drift function improve convergence rate analysis in high-dimensional settings?
• RQ5 What role does dimension suppression play in constructing effective minorization conditions for high-dimensional MCMC algorithms?

Key findings

• The geometric convergence rate of the Markov chain underlying Albert and Chib's algorithm is bounded away from 1 as $n \to \infty$ with $p$ fixed.
• The geometric convergence rate remains bounded away from 1 as $p \to \infty$ with $n$ fixed, indicating robust performance in high-dimensional regimes.
• The paper provides the first computable bounds on the total variation distance to stationarity for this algorithm.
• The refined drift and minorization conditions successfully handle high-dimensional settings by centering the drift and suppressing dimensionality in minorization.
• Asymptotic analysis confirms the stability of the convergence rate across both large sample and high-dimensional covariate scenarios.
• The method enables rigorous, quantitative assessment of MCMC convergence in complex Bayesian models where standard techniques fail.