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The Wishart distribution is the standard conjugate prior for the precision ofthe multivariate Gaussian likelihood, when the mean is known -- while thenormal-Wishart can be used when the mean is also unknown. It is however not soobvious how to assign values to the hyperparameters of these distributions. Inparticular, when forming non-informative limits of these distributions, theshape (or degrees of freedom) parameter of the Wishart must be handled withcare. The intuitive solution of directly interpreting the shape as apseudocount and letting it go to zero, as proposed by some authors, violatesthe restrictions on the shape parameter. We show how to use the scaledKL-divergence between multivariate Gaussians as an energy function to constructWishart and normal-Wishart conjugate priors. When used as informative priors,the salient feature of these distributions is the mode, while the KL scalingfactor serves as the pseudocount. The scale factor can be taken down to thelimit at zero, to form non-informative priors that do not violate therestrictions on the Wishart shape parameter. This limit is non-informative inthe sense that the posterior mode is identical to the maximum likelihoodestimate of the parameters of the Gaussian.

How to use KL-divergence to construct conjugate priors, with well-defined non-informative limits, for the multivariate Gaussian