  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.
