The Matrix-F Prior for Estimating and Testing Covariance Matrices

Abstract

The matrix-$F$ distribution is presented as prior for covariance matrices as an alternative to the conjugate inverted Wishart distribution. A special case of the univariate $F$ distribution for a variance parameter is equivalent to a half-$t$ distribution for a standard deviation, which is becoming increasingly popular in the Bayesian literature. The matrix-$F$ 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-$F$ 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-$F$ distribution. This intrinsic prior is also useful for testing inequality constrained hypotheses on variances. Finally through simulation it is shown that the matrix-variate $F$ distribution has good frequentist properties as prior for the random effects covariance matrix in generalized linear mixed models.