模块化与半模块化贝叶斯推断的后验风险
Posterior risk of modular and semi-modular Bayesian inference
AI总结：
本文首次精确刻画了切断反馈中的偏差-方差权衡，并提出一种新的半模块化后验，证明其在后验风险意义上优于切断后验，从而使切断后验下的点推断不可容许。
摘要：
模块化贝叶斯方法在通过一组耦合子模型（称为模块）指定的模型中进行推断。这些模块通常来源于对不同数据源的建模，或来源于结合不同学科领域的知识。“切断反馈”（Cutting feedback）是一种贝叶斯推断方法，它确保一个模块的误设不会影响其他模块中参数的推断，并产生所谓的切断后验（cut posterior）。然而，在切断后验和标准贝叶斯后验之间进行选择具有挑战性。当误设不严重时，切断反馈会大幅增加后验不确定性，而估计偏差的减少并不大，从而导致偏差-方差权衡。这种权衡促使了半模块化后验（semi-modular posteriors）的产生，它基于一个调节参数在标准后验和切断后验之间进行插值。在这项工作中，我们首次给出了切断反馈中存在的偏差-方差权衡的精确表述，并提出了一种利用该权衡的新的半模块化后验。在一般正则性条件下，我们证明根据一种后验风险概念，该半模块化后验比切断后验更准确。这一结果的一个重要含义是，在切断后验下做出的点推断是不可容许的（inadmissable）。新方法在多个例子中得到了展示。
英文摘要：
Modular Bayesian methods perform inference in models that are specified through a collection of coupled sub-models, known as modules. These modules often arise from modelling different data sources or from combining domain knowledge from different disciplines. ``Cutting feedback'' is a Bayesian inference method that ensures misspecification of one module does not affect inferences for parameters in other modules, and produces what is known as the cut posterior. However, choosing between the cut posterior and the standard Bayesian posterior is challenging. When misspecification is not severe, cutting feedback can greatly increase posterior uncertainty without a large reduction of estimation bias, leading to a bias-variance trade-off. This trade-off motivates semi-modular posteriors, which interpolate between standard and cut posteriors based on a tuning parameter. In this work, we provide the first precise formulation of the bias-variance trade-off that is present in cutting feedback, and we propose a new semi-modular posterior that takes advantage of it. Under general regularity conditions, we prove that this semi-modular posterior is more accurate than the cut posterior according to a notion of posterior risk. An important implication of this result is that point inferences made under the cut posterior are inadmissable. The new method is demonstrated in a number of examples.
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