A1404
Title: Bayesian covariance matrix estimation in high dimensions
Authors: Claudio Busatto - University of Padova (Italy) [presenting]
Mauro Bernardi - University of Padova (Italy)
Manuela Cattelan - University of Padova (Italy)
Alberto Roverato - University of Padova (Italy)
Abstract: A new class of Bayesian priors for estimating covariance matrices in high-dimensional settings where the number of variables can exceed the sample size is proposed. The framework builds upon the inverse Wishart distribution by incorporating off-diagonal shrinkage components. Several existing covariance shrinkage priors are shown to result in improper posterior distributions when the sample covariance matrix is singular, a prevalent issue in high-dimensional problems. In contrast, the proposed RIW priors yield proper posterior distributions under mild conditions, even when the dimension exceeds the sample size. Particular attention is devoted to the CovGLasso inverse Wishart prior, which combines ridge regularization on the precision scale with Laplace shrinkage on off-diagonal covariance entries. The resulting posterior distribution allows for an efficient block Gibbs sampler based on Gaussian scale-mixture representations and latent augmentation techniques. The methodology naturally accommodates adaptive local shrinkage parameters and can be extended to a broad class of continuous shrinkage priors. Theoretical properties of the proposed priors are established, including propriety conditions and induced correlation structures. Overall, the framework provides a computationally tractable and theoretically grounded approach to high-dimensional covariance estimation and graphical modeling.
