We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.

中文翻译：

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更新信念分布的通用框架。


我们提出了一般贝叶斯推理的框架。我们认为，可以对通过损失函数而不是传统的似然函数与观察结果相关的参数进行先验置信分布到后验的有效更新，传统的似然函数作为特殊情况进行恢复。现代应用领域使贝叶斯主义者尝试对真实数据生成机制进行建模变得越来越具有挑战性。例如，当感兴趣的对象是低维的（例如平均值或中位数）时，必须通过整个数据分布的完整模型来实现这一点是很麻烦的。更重要的是，在某些设置中，感兴趣的参数不会直接索引密度函数族，因此学习此类参数的贝叶斯方法目前被认为是有问题的。我们的框架使用损失函数将数据中的信息连接到感兴趣的函数。然后，信念的更新遵循涉及累积损失函数的决策理论方法。重要的是，当已知真实可能性时，该过程与贝叶斯更新一致，但在更一般的设置中提供连贯的主观推理。与其他推理框架的连接被突出显示。