Abstract
The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating) the spectral gaps of practical Monte Carlo Markov chains in statistics has proven to be an extremely difficult and often insurmountable task, especially when these chains move on continuous state spaces. In this paper, a method for accurate estimation of the spectral gap is developed for general state space Markov chains whose operators are non-negative and trace-class. The method is based on the fact that the second largest eigenvalue (and hence the spectral gap) of such operators can be bounded above and below by simple functions of the power sums of the eigenvalues. These power sums often have nice integral representations. A classical Monte Carlo method is proposed to estimate these integrals, and a simple sufficient condition for finite variance is provided. This leads to asymptotically valid confidence intervals for the second largest eigenvalue (and the spectral gap) of the Markov operator. In contrast with previously existing techniques, our method is not based on a near-stationary version of the Markov chain, which, paradoxically, cannot be obtained in a principled manner without bounds on the spectral gap. On the other hand, it can be quite expensive from a computational standpoint. The efficiency of the method is studied both theoretically and empirically.
Highlights
Markov chain Monte Carlo (MCMC) is widely used to estimate intractable integrals that represent expectations with respect to complicated probability distributions
We develop a method of estimating the spectral gaps of Markov operators corresponding to a certain class of data augmentation (DA) algorithms (Tanner and Wong, 1987), and show that the method can be extended to handle a much larger class of reversible MCMC algorithms
We propose a classical Monte Carlo estimator of 1 − δ for DA Markov operators that are trace-class, i.e. compact with summable eigenvalues
Summary
Markov chain Monte Carlo (MCMC) is widely used to estimate intractable integrals that represent expectations with respect to complicated probability distributions. Eigenvalues, Hilbert-Schmidt operator, Markov chain, Monte Carlo want to approximate the integral. We propose a classical Monte Carlo estimator of 1 − δ for DA Markov operators that are trace-class, i.e. compact with summable eigenvalues. If a reversible Monte Carlo Markov chain has a Markov transition density (Mtd), and the corresponding Markov operator is Hilbert-Schmidt, our method can be utilized to estimate its spectral gap. This is because the square of such a Markov operator can be represented as a trace-class DA Markov operator. Further application of the method can be found in Zhang et al (2019)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
