  In this paper we introduce two Bayesian estimators for learning the
parameters of the Gamma distribution. The first algorithm uses a well known
unnormalized conjugate prior for the Gamma shape and the second one uses a
non-linear approximation to the likelihood and a prior on the shape that is
conjugate to the approximated likelihood. In both cases use the Laplace
approximation to compute the required expectations. We perform a theoretical
comparison between maximum like- lihood and the presented Bayesian algorithms
that allow us to provide non-informative parameter values for the priors hyper
parameters. We also provide a numerical comparison using synthetic data. The
introduction of these novel Bayesian estimators open the possibility of
including Gamma distributions into more complex Bayesian structures, e.g.
variational Bayesian mixture models.
