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Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler

Author

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  • Bryant Davis

    (Research & Development, Dallas Cowboys)

  • James P. Hobert

    (University of Florida)

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.

Suggested Citation

  • Bryant Davis & James P. Hobert, 2024. "Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler," Methodology and Computing in Applied Probability, Springer, vol. 26(3), pages 1-13, September.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:3:d:10.1007_s11009-024-10104-y
    DOI: 10.1007/s11009-024-10104-y
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    References listed on IDEAS

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    1. Nicholas G. Polson & James G. Scott & Jesse Windle, 2013. "Bayesian Inference for Logistic Models Using Pólya--Gamma Latent Variables," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1339-1349, December.
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