Bayesian Inference for Polya Inverse Gamma Models

The normalizing constants of these distributions depend on gamma functions whose arguments include shape (gamma, inverse gamma) and concentration (beta, Dirichlet) parameters. Bayesian learning of parameters nested inside the gamma function presents significant technical difficulties, since there is no known conjugate prior distribution. In fact, inferring the shape parameter in the gamma distribution is a long-studied problem in Bayesian inference (Damsleth, 1975; Rossell et al., 2009; Miller, 2018). In this paper, we develop the theoretical and algorithmic foundation of a P olya-inverse Gamma (PIG) data augmentation scheme for fully Bayesian inference of shape and concentration parameters in gamma, inverse gamma, and Dirichlet models, respectively . PIG data augmentation may be utilized to design efficient Markov chain Monte Carlo (MCMC) algorithms in latent Dirichlet allocation (Blei et al., 2003), Beta-negative binomial models (Zhou et al., 2012), and Gamma-Gamma (GaGa) hierarchical models (Rossell et al., 2009).