In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases, statistical measures have been constructed, such as cross validation, information criteria, and marginal likelihood, however, their mathematical properties have not yet been completely clarified when statistical models are under- and over- parametrized. We introduce a place of mathematical theory of Bayesian statistics for unknown uncertainty, which clarifies general properties of cross validation, information criteria, and marginal likelihood, even if an unknown data-generating process is unrealizable by a model or even if the posterior distribution cannot be approximated by any normal distribution. Hence it gives a helpful standpoint for a person who cannot believe in any specific model and prior. This paper consists of three parts. The first is a new result, whereas the second and third are well-known previous results with new experiments. We show there exists a more precise estimator of the generalization loss than leave-one-out cross validation, there exists a more accurate approximation of marginal likelihood than BIC, and the optimal hyperparameters for generalization loss and marginal likelihood are different.

翻译：在统计推断中，不确定性是未知的，所有模型都是错误的。也就是说，构建统计模型和先验分布的人同时意识到这两者都是虚构的候选。为了研究此类情况，人们构建了诸如交叉验证、信息准则和边际似然等统计度量，然而，当统计模型欠参数化或过参数化时，这些度量的数学性质尚未完全明确。我们引入了一个针对未知不确定性的贝叶斯统计数学理论框架，该框架阐明了交叉验证、信息准则和边际似然的一般性质，即使未知的数据生成过程无法被模型实现，或者后验分布不能被任何正态分布近似。因此，它为那些无法相信任何特定模型和先验的人提供了一个有益的视角。本文由三部分组成。第一部分是新的结果，而第二和第三部分是已知的先前结果，并附有新的实验。我们证明存在比留一交叉验证更精确的泛化损失估计量，存在比BIC更准确的边际似然近似方法，并且泛化损失和边际似然的最优超参数是不同的。