## Bernoulli

• Bernoulli
• Volume 25, Number 4B (2019), 3832-3863.

### 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.

#### Article information

Source
Bernoulli, Volume 25, Number 4B (2019), 3832-3863.

Dates
Revised: July 2018
First available in Project Euclid: 25 September 2019

https://projecteuclid.org/euclid.bj/1569398786

Digital Object Identifier
doi:10.3150/19-BEJ1112

Mathematical Reviews number (MathSciNet)
MR4010974

Zentralblatt MATH identifier
07110157

#### Citation

Chakraborty, Saptarshi; Khare, Kshitij. Consistent estimation of the spectrum of trace class Data Augmentation algorithms. Bernoulli 25 (2019), no. 4B, 3832--3863. doi:10.3150/19-BEJ1112. https://projecteuclid.org/euclid.bj/1569398786

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#### Supplemental materials

• Supplement to “Consistent estimation of the spectrum of trace class data augmentation algorithms”. The supplement provides proofs of the theorems and lemmas introduced in this article.