Approximating the Spectral Gap of the P\'olya-Gamma Gibbs Sampler

The self-adjoint, positive Markov operator defined by the P\'olya-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. (2019) to the P\'olya-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.