Multivariate Bayesian Global–Local Shrinkage Methods for Regularisation in the High-Dimensional Linear Model

This paper considers Bayesian regularisation using global–local shrinkage priors in the multivariate general linear model when there are many more explanatory variables than observations. We adopt priors’ structures used extensively in univariate problems (conjugate and non-conjugate with tail behaviour ranging from polynomial to exponential) and consider how the addition of error correlation in the multivariate set-up affects the performance of these priors. Two different datasets (from drug discovery and chemometrics) with many covariates are used for comparison, and these are supplemented by a small simulation study to corroborate the role of error correlation. We find that structural assumptions of the prior distribution on regression coefficients can be more significant than the tail behaviour. In particular, if the structural assumption of conjugacy is used, the performance of the posterior predictive distribution deteriorates relative to non-conjugate choices as the error correlation becomes stronger.