Outlier-robust Bayesian Multivariate Analysis with Correlation-intact Sandwich Mixture

Abstract: Handling outliers is a fundamental challenge in multivariate data analysis, as outliers may distort structures of correlation or conditional independence. Although robust Bayesian inference has been extensively studied for univariate settings, theoretical results ensuring posterior robustness in multivariate models are scarce. We propose a novel scale mixture of multivariate normals called correlation-intact sandwich mixture, where the scale parameters are real-valued and follow the unfolded log-Pareto distribution. Our theoretical results on posterior robustness in multivariate settings emphasizes that the use of a symmetric, super heavy-tailed distribution for the scale parameters is essential in achieving posterior robustness against element-wise contamination. Posterior inference for the proposed model is feasible by an efficient Gibbs sampling algorithm we developed. The superiority of the proposed method is illustrated further in simulation and empirical studies using graphical models and multivariate regression in the presence of complex outlier structures.