Generalised Robust Bayes for Joint Inference of Model and Contamination

2026-07-28

AI Summary

This work addresses the sensitivity of traditional Bayesian inference to contaminated data and the inability of existing robust generalized Bayesian methods to simultaneously quantify contamination levels and identify outliers. The authors propose the Hölder-Bayes framework, which, for the first time within the generalized Bayesian paradigm, jointly infers model parameters and contamination proportion by constructing a generalized posterior via the Hölder divergence, thereby unifying robust inference and anomaly detection. The method provides an interpretable bound on contamination-induced bias and interprets temperature calibration as an affine volume scaling in data space. Leveraging posterior influence functions, Bernstein–von Mises approximations, and uncertainty propagation, it achieves accurate estimation of contamination proportions, robust parameter inference, and threshold-free, uncertainty-aware anomaly detection through Frequency-of-Detection scores.

Abstract

Generalised Bayesian inference (GBI) has emerged as a compelling robust alternative to standard Bayesian inference, mitigating sensitivity to data contamination by replacing the log-likelihood with a robust loss or divergence. However, existing robust GBI frameworks typically provide only qualitative robustness: while they can make posterior inference less sensitive to contamination, they lack an intrinsic mechanism to quantify the contamination proportion or identify anomalous observations. This paper introduces Hölder-Bayes, a GBI framework for joint inference of the model parameter and the contamination proportion. We construct a generalised joint posterior over both model and contamination parameter by applying the Hölder divergence to a scaled model density. Theoretically, we establish global bias-robustness via the uniform boundedness of the posterior influence function, derive a finite-sample excess-risk bound, and prove a Bernstein--von Mises approximation together with interpretable contamination-induced bias bounds under a heavy-contamination regime. We further show that, for the Hölder posterior, temperature calibration admits a direct interpretation as affine volume scaling of the data space. The resulting posterior yields a self-contained probabilistic mechanism for outlier detection: posterior uncertainty in both the model parameter and the contamination proportion is propagated to observation-level Frequency-of-Detection scores, without requiring an external anomaly-score threshold. Empirical evaluations demonstrate that Hölder-Bayes provides robust parameter inference, contamination-level recovery, and uncertainty-aware outlier detection.

Problem

• robust Bayesian inference
• data contamination
• contamination proportion
• outlier detection
• generalised Bayesian inference

Innovation

• Hölder-Bayes
• generalised Bayesian inference
• contamination proportion
• outlier detection
• posterior robustness