Robustness of Bayesian ordinal response model against outliers via divergence approach

TL;DR

This paper investigates the vulnerability of Bayesian ordinal response models to outliers and introduces robust divergence-based posteriors to improve inference reliability in the presence of outliers, supported by theoretical proofs and numerical experiments.

Contribution

It proves the lack of posterior robustness in standard Bayesian ordinal models and proposes new robust divergence-based methods with algorithms and empirical validation.

Findings

Standard Bayesian ordinal models are not robust to outliers.

Proposed divergence-based posteriors improve robustness against outliers.

Numerical experiments demonstrate superior performance of the proposed methods.

Abstract

Ordinal response model is a popular and commonly used regression for ordered categorical data in a wide range of fields such as medicine and social sciences. However, it is empirically known that the existence of ``outliers'', combinations of the ordered categorical response and covariates that are heterogeneous compared to other pairs, makes the inference with the ordinal response model unreliable. In this article, we prove that the posterior distribution in the ordinal response model does not satisfy the posterior robustness with any link functions, i.e., the posterior cannot ignore the influence of large outliers. Furthermore, to achieve robust Bayesian inference in the ordinal response model, this article defines general posteriors in the ordinal response model with two robust divergences (the density-power and $\gamma$-divergences) based on the framework of the general posterior inference. We also provide an algorithm for generating posterior samples from the proposed posteriors. The robustness of the proposed methods against outliers is clarified from the posterior robustness and the index of robustness based on the Fisher-Rao metric. Through numerical experiments on artificial data and two real datasets, we show that the proposed methods perform better than the ordinary bayesian methods with and without outliers in the data for various link functions.