  The Wishart distribution is the standard conjugate prior for the precision of
the multivariate Gaussian likelihood, when the mean is known -- while the
normal-Wishart can be used when the mean is also unknown. It is however not so
obvious how to assign values to the hyperparameters of these distributions. In
particular, when forming non-informative limits of these distributions, the
shape (or degrees of freedom) parameter of the Wishart must be handled with
care. The intuitive solution of directly interpreting the shape as a
pseudocount and letting it go to zero, as proposed by some authors, violates
the restrictions on the shape parameter. We show how to use the scaled
KL-divergence between multivariate Gaussians as an energy function to construct
Wishart and normal-Wishart conjugate priors. When used as informative priors,
the salient feature of these distributions is the mode, while the KL scaling
factor serves as the pseudocount. The scale factor can be taken down to the
limit at zero, to form non-informative priors that do not violate the
restrictions on the Wishart shape parameter. This limit is non-informative in
the sense that the posterior mode is identical to the maximum likelihood
estimate of the parameters of the Gaussian.
