We introduce a novel Information Criterion (IC), termed Learning under Singularity (LS), designed to enhance the functionality of the Widely Applicable Bayes Information Criterion (WBIC) and the Singular Bayesian Information Criterion (sBIC). LS is effective without regularity constraints and demonstrates stability. Watanabe defined a statistical model or a learning machine as regular if the mapping from a parameter to a probability distribution is one-to-one and its Fisher information matrix is positive definite. In contrast, models not meeting these conditions are termed singular. Over the past decade, several information criteria for singular cases have been proposed, including WBIC and sBIC. WBIC is applicable in non-regular scenarios but faces challenges with large sample sizes and redundant estimation of known learning coefficients. Conversely, sBIC is limited in its broader application due to its dependence on maximum likelihood estimates. LS addresses these limitations by enhancing the utility of both WBIC and sBIC. It incorporates the empirical loss from the Widely Applicable Information Criterion (WAIC) to represent the goodness of fit to the statistical model, along with a penalty term similar to that of sBIC. This approach offers a flexible and robust method for model selection, free from regularity constraints.

我们引入了一种新颖的信息准则（IC），称为孤立学习（LS），用来增强广泛适用贝叶斯信息准则（WBIC）和奇异贝叶斯信息准则（sBIC）的功能。LS在没有正则约束的情况下有效，并表现出稳定性，通过将广泛适用信息准则（WAIC）的经验损失与sBIC类似的惩罚项相结合，LS提供了一个灵活且鲁棒的模型选择方法，不受正则约束限制。

在奇异点下的学习: 一种改进 WBIC 和 sBIC 的信息准则