Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler
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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-λ_*, where λ_* ∈ [0,1) is the second largest eigenvalue. A method of constructing an asymptotically valid confidence interval for an upper bound on λ_* is developed by adapting the classical Monte Carlo technique of Qin et al. (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.
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