Estimating the spectral gap of a trace-class Markov operator

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.