Robust Bayesian precision matrix estimation with a regularized horseshoe prior

Abstract

Estimating high-dimensional precision matrices in Gaussian graphical models is challenging due to sensitivity to outliers and limitations of uniform shrinkage methods. We propose a robust Bayesian approach based on a sparse Cholesky factorization, combining the \(\gamma\)-divergence for outlier resistance with a regularized horseshoe prior to enable adaptive shrinkage. Estimation is performed via MAP optimization using the Adam algorithm, with uncertainty quantified through a Laplace approximation. Theoretical properties including posterior propriety, existence of the MAP estimator, and robustness are established. Simulation results show improved performance in terms of Frobenius norm and root mean squared error under data contamination. An application to gene expression data illustrates the method’s ability to identify potentially meaningful association patterns. The proposed framework provides a computationally tractable approach to robust precision matrix estimation in high-dimensional settings.