Geometric ergodicity of Gibbs samplers for the Horseshoe and its regularized variants

Suman Bhattacharya, Kshitij Khare, Subhadip Pal

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

The Horseshoe is a widely used and popular continuous shrinkage prior for high-dimensional Bayesian linear regression. Recently, regularized versions of the Horseshoe prior have also been introduced in the literature. Various Gibbs sampling Markov chains have been developed in the literature to generate approximate samples from the corresponding intractable posterior densities. Establishing geometric ergodicity of these Markov chains provides crucial technical justification for the accuracy of asymptotic standard errors for Markov chain based estimates of posterior quantities. In this paper, we establish geometric ergodicity for various Gibbs samplers corresponding to the Horseshoe prior and its regularized variants in the context of linear regression. First, we establish geometric ergodicity of a Gibbs sampler for the original Horseshoe posterior under strictly weaker conditions than existing analyses in the literature. Second, we consider the regularized Horseshoe prior introduced in [18], and prove geometric ergod-icity for a Gibbs sampling Markov chain to sample from the corresponding posterior without any truncation constraint on the global and local shrinkage parameters. Finally, we consider a variant of this regularized Horseshoe prior introduced in [15], and again establish geometric ergodicity for a Gibbs sampling Markov chain to sample from the corresponding posterior.

Original languageEnglish
Pages (from-to)1-57
Number of pages57
JournalElectronic Journal of Statistics
Volume16
Issue number1
DOIs
Publication statusPublished - 2022
Externally publishedYes

Keywords

  • geometric ergodicity
  • high-dimensional linear regression
  • Horseshoe prior
  • Markov chain Monte Carlo

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint

Dive into the research topics of 'Geometric ergodicity of Gibbs samplers for the Horseshoe and its regularized variants'. Together they form a unique fingerprint.

Cite this