Title: Fast and accurate approximation of the full conditional for gamma shape parameters,
Abstract: The gamma distribution arises frequently in Bayesian models, but there is not
an easy-to-use conjugate prior for the shape parameter of a gamma. This
inconvenience is usually dealt with by using either Metropolis-Hastings moves,
rejection sampling methods, or numerical integration. However, in models with a
large number of shape parameters, these existing methods are slower or more
complicated than one would like, making them burdensome in practice. It turns
out that the full conditional distribution of the gamma shape parameter is well
approximated by a gamma distribution, even for small sample sizes, when the
prior on the shape parameter is also a gamma distribution. This article
introduces a quick and easy algorithm for finding a gamma distribution that
approximates the full conditional distribution of the shape parameter. We
empirically demonstrate the speed and accuracy of the approximation across a
wide range of conditions. If exactness is required, the approximation can be
used as a proposal distribution for Metropolis-Hastings.