Marketing Papers

Document Type

Technical Report

Date of this Version

2002

Publication Source

Journal of Computational and Graphical Statistics

Volume

11

Issue

1

Start Page

189

Last Page

201

DOI

10.1198/106186002317375677

Abstract

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.

Copyright/Permission Statement

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.

Keywords

beta-prime distribution, empirical bayes methods, Pearson type VI distribution

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Date Posted: 15 June 2018

This document has been peer reviewed.