Consistent estimation of the spectrum of trace class data augmentation algorithms

TL;DR

This paper introduces a new method for consistently estimating the entire spectrum of Markov chains from data augmentation algorithms, addressing the challenge of intractable transition densities in practical applications.

Contribution

The authors develop a novel technique to estimate the full spectrum of Markov chains with intractable transition densities, expanding the tools for analyzing convergence in complex models.

Findings

Method successfully estimates the spectrum from real and simulated data.

Applicable to a broad class of Markov chains in statistical applications.

Addresses limitations of existing eigenvalue estimation techniques.

Abstract

Markov chain Monte Carlo is widely used in a variety of scientific applications to generate approximate samples from intractable distributions. A thorough understanding of the convergence and mixing properties of these Markov chains can be obtained by studying the spectrum of the associated Markov operator. While several methods to bound/estimate the second largest eigenvalue are available in the literature, very few general techniques for consistent estimation of the entire spectrum have been proposed. Existing methods for this purpose require the Markov transition density to be available in closed form, which is often not true in practice, especially in modern statistical applications. In this paper, we propose a novel method to consistently estimate the entire spectrum of a general class of Markov chains arising from a popular and widely used statistical approach known as Data Augmentation. The transition densities of these Markov chains can often only be expressed as intractable integrals. We illustrate the applicability of our method using real and simulated data.