Geometric ergodicity of Polya-Gamma Gibbs sampler for Bayesian logistic regression with a flat prior

TL;DR

This paper proves that the Polya-Gamma Gibbs sampler for Bayesian logistic regression with a flat prior is geometrically ergodic, ensuring reliable statistical inference through CLTs and standard error calculations.

Contribution

It establishes the geometric ergodicity of the Polya-Gamma Gibbs sampler, a key theoretical property for Bayesian logistic regression with improper priors.

Findings

Proves geometric ergodicity of the Markov chain

Ensures validity of CLTs for posterior estimates

Facilitates standard error computation in Bayesian analysis

Abstract

The logistic regression model is the most popular model for analyzing binary data. In the absence of any prior information, an improper flat prior is often used for the regression coefficients in Bayesian logistic regression models. The resulting intractable posterior density can be explored by running Polson et al.'s (2013) data augmentation (DA) algorithm. In this paper, we establish that the Markov chain underlying Polson et al.'s (2013) DA algorithm is geometrically ergodic. Proving this theoretical result is practically important as it ensures the existence of central limit theorems (CLTs) for sample averages under a finite second moment condition. The CLT in turn allows users of the DA algorithm to calculate standard errors for posterior estimates.