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：本文提出一种新颖的信息准则（IC），称为“奇异性下的学习”（LS），旨在增强广泛适用贝叶斯信息准则（WBIC）与奇异贝叶斯信息准则（sBIC）的功能。LS在无正则性约束条件下依然有效，且表现出稳定性。Watanabe将统计模型或学习机定义为正则的，当且仅当从参数到概率分布的映射是一一映射，且其Fisher信息矩阵正定。反之，不满足这些条件的模型则称为奇异模型。过去十年间，已提出若干针对奇异情形的信息准则，包括WBIC与sBIC。WBIC适用于非正则场景，但在大样本量下以及已知学习系数的冗余估计中面临挑战。相反，sBIC因依赖于极大似然估计而限制了其更广泛的应用。LS通过增强WBIC与sBIC的实用性来克服这些局限。它融合了广泛适用信息准则（WAIC）中表示统计模型拟合优度的经验损失，并结合了类似sBIC的惩罚项。该方法为模型选择提供了一种灵活且稳健的途径，且无需正则性约束。