Estimation of a covariance matrix $\sum$ is a notoriously difficult problem; the standard unbiased estimator can be substantially suboptimal. We approach the problem from a noninformative prior Bayesian perspective, developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting Bayes estimators. These expressions involve the computation of high-dimensional posterior expectations, which is done using a recent Markov chain simulation tool, the hit-and-run sampler. Frequentist risk comparisons with previously suggested estimators are also given, and determination of the accuracy of the estimators is addressed.
"Estimation of a Covariance Matrix Using the Reference Prior." Ann. Statist. 22 (3) 1195 - 1211, September, 1994. https://doi.org/10.1214/aos/1176325625