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.

翻译：我们提出了一种新颖的信息准则（IC），称为奇异点下的学习（LS），其旨在增强广泛适用的贝叶斯信息准则（WBIC）和奇异贝叶斯信息准则（sBIC）的功能性。LS在无正则性约束条件下有效，并表现出稳定性。Watanabe将统计模型或学习机定义为正则的，如果从参数到概率分布的映射是一一对应的且其Fisher信息矩阵为正定。反之，不满足这些条件的模型被称为奇异的。在过去十年中，针对奇异情况提出了若干信息准则，包括WBIC和sBIC。WBIC适用于非正则场景，但面临大样本量和对已知学习系数的冗余估计带来的挑战。相比之下，sBIC因其对极大似然估计的依赖而在更广泛的应用中受限。LS通过增强WBIC和sBIC的实用性来克服这些局限性。它融合了广泛适用的信息准则（WAIC）中的经验损失以表示对统计模型的拟合优度，并结合了类似sBIC的惩罚项。该方法提供了一种灵活且稳健的模型选择方式，不受正则性约束的限制。