A New Bayesian Huberised Regularisation and Beyond

TL;DR

This paper introduces a novel Bayesian Huberised regularisation method for robust regression that handles asymmetry and high-dimensional data, providing full probabilistic uncertainty quantification and demonstrating superior performance in simulations and real data.

Contribution

It proposes a new Huberised asymmetric loss function with a scale-mixture of normals distribution and develops a Bayesian regularisation approach, including a Bayesian Huberised quantile regression.

Findings

Demonstrates robustness and effectiveness in simulations.

Shows improved uncertainty quantification over frequentist methods.

Validates models with real data analysis.

Abstract

Robust regression has attracted a great amount of attention in the literature recently, particularly for taking asymmetricity into account simultaneously and for high-dimensional analysis. However, the majority of research on the topics falls in frequentist approaches, which are not capable of full probabilistic uncertainty quantification. This paper first proposes a new Huberised-type of asymmetric loss function and its corresponding probability distribution which is shown to have the scale-mixture of normals. Then we introduce a new Bayesian Huberised regularisation for robust regression. A by-product of the research is that a new Bayesian Huberised regularised quantile regression is also derived. We further present their theoretical posterior properties. The robustness and effectiveness of the proposed models are demonstrated in the simulation studies and the real data analysis.