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The Pólya-gamma Gibbs sampler for Bayesian logistic regression is uniformly ergodic. (English) Zbl 1349.60123
Summary: One of the most widely used data augmentation algorithms is J. H. Albert and S. Chib’s [J. Am. Stat. Assoc. 88, No. 422, 669–679 (1993; Zbl 0774.62031)] algorithm for Bayesian probit regression. N. G. Polson et al. [J. Am. Stat. Assoc. 108, No. 504, 1339–1349 (2013; Zbl 1283.62055)] recently introduced an analogous algorithm for Bayesian logistic regression. The main difference between the two is that Albert and Chib’s [loc. cit.] truncated normals are replaced by so-called Polya-Gamma random variables. In this note, we establish that the Markov chain underlying Polson et al.’s [loc. cit.] 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.

60J22 Computational methods in Markov chains
62J12 Generalized linear models (logistic models)
62F15 Bayesian inference
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