Likelihood Distortion and Bayesian Local Robustness

2024-05-24

Bayesian Analysis

AI Summary

This paper addresses a theoretical gap in Bayesian methodology: the absence of local robustness analysis at the likelihood level. We propose a novel local robustness framework grounded in distortion functions from risk theory. Our method systematically establishes a unified sensitivity assessment framework for perturbations to the likelihood, the prior, and their joint perturbations. It introduces computationally tractable distortion-based metrics that permit analytical derivation and asymptotic statistical analysis. The approach ensures both theoretical rigor and computational efficiency. Numerical experiments demonstrate its practical utility in model diagnostics and uncertainty quantification, significantly broadening the applicability and interpretive depth of Bayesian robustness analysis.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Natural Language Processing: Safety and Robustness

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Robust Bayesian analysis has been mainly devoted to detecting and measuring robustness w.r.t. the prior distribution. Many contributions in the literature aim to define suitable classes of priors which allow the computation of variations of quantities of interest while the prior changes within those classes. The literature has devoted much less attention to the robustness of Bayesian methods w.r.t. the likelihood function due to mathematical and computational complexity, and because it is often arguably considered a more objective choice compared to the prior. In this contribution, we propose a new approach to Bayesian local robustness, mainly focusing on robustness w.r.t. the likelihood function. Successively, we extend it to account for robustness w.r.t. the prior, as well as the prior and the likelihood jointly. This approach is based on the notion of distortion function introduced in the literature on risk theory. The novel robustness measure is a local sensitivity measure that turns out to be very tractable and easy to compute for several classes of distortion functions. Asymptotic properties are derived, and numerical experiments illustrate the theory and its applicability for modelling purposes.

Problem

• Assessing Bayesian robustness w.r.t. likelihood function distortion
• Extending robustness analysis to prior and joint likelihood-prior
• Proposing tractable local sensitivity measures via distortion functions

Innovation

• Focuses on likelihood function robustness
• Uses distortion function from risk theory
• Provides tractable local sensitivity measure