Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler

Bryant Davis () and James P. Hobert ()
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Bryant Davis: Research & Development, Dallas Cowboys
James P. Hobert: University of Florida

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 3, 1-13

Abstract: Abstract 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 $$1-\lambda _*$$ 1 - λ ∗ , where $$\lambda _* \in [0,1)$$ λ ∗ ∈ [ 0 , 1 ) is the second largest eigenvalue. A method of constructing an asymptotically valid confidence interval for an upper bound on $$\lambda _*$$ λ ∗ 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.

Keywords: Bayesian logistic regression; Compact operator; Geometric ergodicity; Markov chain Monte Carlo; Markov operator; Trace-class operator; Uniform ergodicity (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10104-y

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