Bayesian Inference for the Negative Binomial Distribution
via Polynomial Expansions
Eric T. Bradlow
Bruce G.S. Hardie
Peter S. Fader
Journal of Computational and Graphical Statistics
Volume 11, Number 1
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 paper,
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.