ABSTRACT In the context of predicting future claims, a fully Bayesian analysis – one that specifies a statistical model, prior distribution, and updates using Bayes's formula – is often viewed as the gold-standard, while Bühlmann's credibility estimator serves as a simple approximation. But those desirable properties that give the Bayesian solution its elevated status depend critically on the posited model being correctly specified. Here we investigate the asymptotic behavior of Bayesian posterior distributions under a misspecified model, and our conclusion is that misspecification bias generally has damaging effects that can lead to inaccurate inference and prediction. The credibility estimator, on the other hand, is not sensitive at all to model misspecification, giving it an advantage over the Bayesian solution in those practically relevant cases where the model is uncertain. This begs the question: does robustness to model misspecification require that we abandon uncertainty quantification based on a posterior distribution? Our answer to this question is No, and we offer an alternative Gibbs posterior construction. Furthermore, we argue that this Gibbs perspective provides a new characterization of Bühlmann's credibility estimator.

中文翻译：

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模型错误指定、贝叶斯与可信度估计和吉布斯后验

摘要 在预测未来索赔的背景下，完全贝叶斯分析——一种使用贝叶斯公式指定统计模型、先验分布和更新的分析——通常被视为黄金标准，而 Bühlmann 的可信度估计则作为一个简单的近似值。但是，那些赋予贝叶斯解决方案提升地位的理想属性关键取决于正确指定的假设模型。在这里，我们研究了错误指定模型下贝叶斯后验分布的渐近行为，我们的结论是错误指定偏差通常具有破坏性影响，可能导致推理和预测不准确。另一方面，可信度估计器对模型错误指定完全不敏感，在模型不确定的实际相关情况下，使其优于贝叶斯解决方案。这就引出了一个问题：模型错误指定的鲁棒性是否要求我们放弃基于后验分布的不确定性量化？我们对这个问题的回答是否定的，我们提供了另一种 Gibbs 后验构造。此外，我们认为这种 Gibbs 观点提供了 Bühlmann 可信度估计量的新特征。