In statistical inference, the uncertainty is unknown and all models are wrong. That is, those who create statistical models and priors are aware that both are hypothetical candidates at the same time. Statistical measures such as cross-validation, information criteria, and marginal likelihood have been constructed to study such cases, but their mathematical properties have not yet been fully elucidated.

    Introduces the place of mathematical theory of Bayesian statistics for unknown uncertainties. It reveals general properties of cross-validation, information criteria, and marginal likelihood, even when the unknown data-generating process cannot be realized by the model, or even when the posterior distribution cannot be realized. Approximated by an arbitrary normal distribution. It therefore provides a helpful standpoint for those who cannot believe in any particular model or precedent.

    This document is divided into three parts. The first is a new result, the second and his third are well-known previous results from new experiments. There are more accurate estimators of generalized loss than leave-one-out cross-validation, more accurate approximations of marginal likelihood than BIC, and different optimal hyperparameters for generalized loss and marginal likelihood. indicates

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