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

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, Scott and Windle’s (2013) data augmentation (DA) algorithm. In this paper, we establish that the Markov chain underlying Polson, Scott and Windle’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.