Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler

The self-adjoint, positive Markov operator defined by the Pólya-Gamma Gibbs sampler (under a proper normal prior) is shown to be trace-class, which implies that all non-zero elements of its spectrum are eigenvalues. Consequently, the spectral gap is \(1-\lambda _*\), where \(\lambda _* \in [0,1)\) is the second largest eigenvalue. A method of constructing an asymptotically valid confidence interval for an upper bound on \(\lambda _*\) is developed by adapting the classical Monte Carlo technique of Qin et al. (Electron J Stat 13:1790–1812, 2019) to the Pólya-Gamma Gibbs sampler. The results are illustrated using the German credit data. It is also shown that, in general, uniform ergodicity does not imply the trace-class property, nor does the trace-class property imply uniform ergodicity.