The matrix- distribution is presented as prior for covariance matrices as an alternative to the conjugate inverted Wishart distribution. A special case of the univariate distribution for a variance parameter is equivalent to a half- distribution for a standard deviation, which is becoming increasingly popular in the Bayesian literature. The matrix- distribution can be conveniently modeled as a Wishart mixture of Wishart or inverse Wishart distributions, which allows straightforward implementation in a Gibbs sampler. By mixing the covariance matrix of a multivariate normal distribution with a matrix- distribution, a multivariate horseshoe type prior is obtained which is useful for modeling sparse signals. Furthermore, it is shown that the intrinsic prior for testing covariance matrices in non-hierarchical models has a matrix- distribution. This intrinsic prior is also useful for testing inequality constrained hypotheses on variances. Finally through simulation it is shown that the matrix-variate distribution has good frequentist properties as prior for the random effects covariance matrix in generalized linear mixed models.
"The Matrix- Prior for Estimating and Testing Covariance Matrices." Bayesian Anal. 13 (4) 1193 - 1214, December 2018. https://doi.org/10.1214/17-BA1092