Keywords: Wishart distribution, shrinkage inverse Wishart distributon, Jeffreys prior, covariance matrix, low rank learning
Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this talk, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors -- both subjective and objective -- that do not ``force eigenvalues apart," which is a criticism of inverse Wishart and Jeffreys priors. Extensive comparison of these `shrinkage priors' with inverse Wishart and Jeffreys priors is undertaken, with the new priors seeming to have considerably better performance. A number of curious facts about the new priors are also observed, such as that the posterior distribution will be proper with just three vector observations from the multivariate normal distribution -- regardless of the dimension of the covariance matrix -- and that useful inference about features of the covariance matrix can be possible. Finally, a new MCMC algorithm is developed for this class of priors and is shown to be computationally effective for matrices of up to 100 dimensions.