Date of this Version
Journal of Computational and Graphical Statistics
To date, Bayesian inferences for the negative binomial distribution (NBD) have relied on computationally intensive numerical methods (e.g., Markov chain Monte Carlo) as it is thought that the posterior densities of interest are not amenable to closed-form integration. In this article, we present a “closed-form” solution to the Bayesian inference problem for the NBD that can be written as a sum of polynomial terms. The key insight is to approximate the ratio of two gamma functions using a polynomial expansion, which then allows for the use of a conjugate prior. Given this approximation, we arrive at closed-form expressions for the moments of both the marginal posterior densities and the predictive distribution by integrating the terms of the polynomial expansion in turn (now feasible due to conjugacy). We demonstrate via a large-scale simulation that this approach is very accurate and that the corresponding gains in computing time are quite substantial. Furthermore, even in cases where the computing gains are more modest our approach provides a method for obtaining starting values for other algorithms, and a method for data exploration.
This is an Accepted Manuscript of an article published by Taylor & Francis in the Journal of Computational and Graphical Statistics on January 1, 2002, available online: http://dx.doi.org/10.1198/106186002317375677.
beta-prime distribution, empirical bayes methods, Pearson type VI distribution
Bradlow, E. T., Hardie, B. G., & Fader, P. S. (2002). Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions. Journal of Computational and Graphical Statistics, 11 (1), 189-201. http://dx.doi.org/10.1198/106186002317375677
Date Posted: 15 June 2018
This document has been peer reviewed.