Bayesian Inference for Logistic Models Using Pólya–Gamma Latent Variables

Abstract

We propose a new data-augmentation strategy for fully Bayesian inference in models with binomial likelihoods. The approach appeals to a new class of Pólya–Gamma distributions, which are constructed in detail. A variety of examples are presented to show the versatility of the method, including logistic regression, negative binomial regression, nonlinear mixed-effect models, and spatial models for count data. In each case, our data-augmentation strategy leads to simple, effective methods for posterior inference that (1) circumvent the need for analytic approximations, numerical integration, or Metropolis–Hastings; and (2) outperform other known data-augmentation strategies, both in ease of use and in computational efficiency. All methods, including an efficient sampler for the Pólya–Gamma distribution, are implemented in the R package BayesLogit. Supplementary materials for this article are available online.

Keywords:

• Bayesian methods
• Data augmentation
• Logistic regression
• Negative binomial regression
• Pólya–Gamma distribution

Acknowledgments

The authors thank Hee Min Choi and Jim Hobert for sharing an early draft of their article on the uniform ergodicity of the Pólya–Gamma Gibbs sampler. They also thank two anonymous referees, the associate editor, and the editor of the Journal of the American Statistical Association, whose many insights and helpful suggestions have improved the article. The second author acknowledges the support of a CAREER grant from the U.S. National Science Foundation (DMS-1255187).