  We propose a new data-augmentation strategy for fully Bayesian inference in
models with binomial likelihoods. The approach appeals to a new class of
Polya-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-effects
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 Polya-Gamma distribution, are
implemented in the R package BayesLogit.
  In the technical supplement appended to the end of the paper, we provide
further details regarding the generation of Polya-Gamma random variables; the
empirical benchmarks reported in the main manuscript; and the extension of the
basic data-augmentation framework to contingency tables and multinomial
outcomes.
