Consistent estimation of the spectrum of trace class data augmentation algorithms

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.

arXiv comment43 pages (including Appendix), 3 figures; final version

References in corpus (8)

• A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants
• Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions
• Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling
• Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors
• Some remarks on MCMC estimation of spectra of integral operators
• Scalable Bayesian shrinkage and uncertainty quantification in high-dimensional regression
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• Gibbs Sampling, Exponential Families and Orthogonal Polynomials