AI 中文总结
本文针对现有广义贝叶斯推断框架缺乏量化污染比例及识别异常观测机制的问题，提出Hölder-Bayes框架，通过构建广义联合后验实现模型参数与污染比例联合推断，经理论推导和实证评估，具备稳健参数推断等能力。
AI 中文摘要
广义贝叶斯推断（GBI）已成为标准贝叶斯推断的一种有吸引力的稳健替代方法，通过用稳健损失或散度代替对数似然来减轻对数据污染的敏感性。然而，现有的稳健GBI框架通常只提供定性稳健性。本文介绍了Hölder-Bayes，一种用于模型参数和污染比例联合推断的GBI框架。通过对缩放后的模型密度应用Hölder散度，构建了模型和污染参数的广义联合后验。理论上，通过后验影响函数的一致有界性建立了全局偏差稳健性，推导了有限样本超额风险界，并在重污染情况下证明了伯恩斯坦 - 冯·米塞斯近似以及可解释的污染诱导偏差界。还表明温度校准可直接解释为数据空间的仿射体积缩放。实证评估表明Hölder-Bayes提供了稳健的参数推断、污染水平恢复和不确定性感知的异常检测。
英文摘要
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.
Comments50 pages, 3 figures
