Data Augementation with Polya Inverse Gamma
This provides a scalable solution for statisticians and machine learning practitioners working with models like Multinomial-Dirichlet and Poisson-Gamma, though it is incremental as it builds on existing data augmentation theory.
The authors tackled the challenge of inference and prediction in statistical models containing gamma functions without natural conjugate priors by developing a data augmentation scheme using Pólya Inverse Gamma distributions, enabling scalable EM and MCMC algorithms for applications like gamma shape inference and negative binomial regression.
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 Pólya 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.