The matrix-F prior for estimating and testing covariance matrices. (English) Zbl 1407.62080

Summary: 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.


62F15 Bayesian inference
62H12 Estimation in multivariate analysis
62H10 Multivariate distribution of statistics
62J12 Generalized linear models (logistic models)


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