One of the most widely used data augmentation algorithms is Albert and Chib’s (1993) algorithm for Bayesian probit regression. Polson, Scott, and Windle (2013) recently introduced an analogous algorithm for Bayesian logistic regression. The main difference between the two is that Albert and Chib’s (1993) truncated normals are replaced by so-called Polya-Gamma random variables. In this note, we establish that the Markov chain underlying Polson, Scott, and Windle’s (2013) algorithm is uniformly ergodic. This theoretical result has important practical benefits. In particular, it guarantees the existence of central limit theorems that can be used to make an informed decision about how long the simulation should be run.
"The Polya-Gamma Gibbs sampler for Bayesian logistic regression is uniformly ergodic." Electron. J. Statist. 7 2054 - 2064, 2013. https://doi.org/10.1214/13-EJS837