Gamma Processes, Stick-Breaking, and Variational Inference

While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process, the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric prior in its own right. Current inference schemes for models involving the gamma process are restricted to MCMC-based methods, which limits their scalability. In this paper, we present a variational inference framework for models involving gamma process priors. Our approach is based on a novel stick-breaking constructive definition of the gamma process. We prove correctness of this stick-breaking process by using the characterization of the gamma process as a completely random measure (CRM), and we explicitly derive the rate measure of our construction using Poisson process machinery. We also derive error bounds on the truncation of the infinite process required for variational inference, similar to the truncation analyses for other nonparametric models based on the Dirichlet and beta processes. Our representation is then used to derive a variational inference algorithm for a particular Bayesian nonparametric latent structure formulation known as the infinite Gamma-Poisson model, where the latent variables are drawn from a gamma process prior with Poisson likelihoods. Finally, we present results for our algorithms on nonnegative matrix factorization tasks on document corpora, and show that we compare favorably to both sampling-based techniques and variational approaches based on beta-Bernoulli priors

Estimate of revenues from the value added tax in the Republic of Croatia

This Occasional Paper is part of a research project undertaken by the Institute of Public Finance and financed by the Ministry of Finance of the Republic of Croatia. The research project and the presented paper are published in Croatian in the Institute’s journal “Financijska praksa”, Volume 20, Number 2 (August 1996). Part I of the paper is written by Danijela Kuliš (Institute of Public Finance, Zagreb), and Part II is written by Žarko Miljenović (then, State Bureau for Macroeconomic Analyses and Forecasts, Zagreb; and now Zagrebačka banka, Zagreb). The project’s lead researcher was Dr. Katarina Ott (Institute of Public Finance, Zagreb)