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

Valentina Mameli, Debora Slanzi, Jim E. Griffin and Philip J. Brown ()
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Valentina Mameli: Department of Economics and Statistics, University of Udine, 33100 Udine, Italy
Debora Slanzi: Venice School of Management, Ca’ Foscari University of Venice, 30121 Venice, Italy
Jim E. Griffin: Department of Statistical Science, University College London, London WC1E 6BT, UK
Philip J. Brown: Department of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NZ, UK

Mathematics, 2025, vol. 13, issue 11, 1-21

Abstract: 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.

Keywords: seemingly unrelated multivariate normal regression; structured priors; global–local priors; exponential-tailed and polynomial-tailed priors; drug discovery; chemometrics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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