  We use the theory of normal variance-mean mixtures to derive a data
augmentation scheme for models that include gamma functions. Our methodology
applies to many situations in statistics and machine learning, including
Multinomial-Dirichlet distributions, Negative binomial regression,
Poisson-Gamma hierarchical models, Extreme value models, to name but a few. All
of those models include a gamma function which does not admit a natural
conjugate prior distribution providing a significant challenge to inference and
prediction. To provide a data augmentation strategy, we construct and develop
the theory of the class of P\'olya Inverse Gamma distributions. This allows
scalable EM and MCMC algorithms to be developed. We illustrate our methodology
on a number of examples, including gamma shape inference, negative binomial
regression and Dirichlet allocation. Finally, we conclude with directions for
future research.
