  In this paper we introduce five different algorithms based on method of
moments, maximum likelihood and full Bayesian estimation for learning the
parameters of the Inverse Gamma distribution. We also provide an expression for
the KL divergence for Inverse Gamma distributions which allows us to quantify
the estimation accuracy of each of the algorithms. All the presented algorithms
are novel. The most relevant novelties include the first conjugate prior for
the Inverse Gamma shape parameter which allows analytical Bayesian inference,
and two very fast algorithms, a maximum likelihood and a Bayesian one, both
based on likelihood approximation. In order to compute expectations under the
proposed distributions we use the Laplace approximation. The introduction of
these novel Bayesian estimators opens the possibility of including Inverse
Gamma distributions into more complex Bayesian structures, e.g. variational
Bayesian mixture models. The algorithms introduced in this paper are
computationally compared using synthetic data and interesting relationships
between the maximum likelihood and the Bayesian approaches are derived.
