# Data Augementation with Polya Inverse Gamma

## TL;DR

This paper introduces a novel data augmentation method using Pólya Inverse Gamma distributions to improve inference in models involving gamma functions, applicable to various statistical and machine learning models.

## Contribution

It develops the theory of Pólya Inverse Gamma distributions and demonstrates scalable EM and MCMC algorithms for models with gamma functions, addressing a key inference challenge.

## Key findings

- Effective data augmentation for gamma-related models
- Scalable algorithms for inference in complex models
- Successful application to negative binomial regression and Dirichlet allocation

## Abstract

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.