The learning coefficient plays a crucial role in analyzing the performance of information criteria, such as the Widely Applicable Information Criterion (WAIC) and the Widely Applicable Bayesian Information Criterion (WBIC), which Sumio Watanabe developed to assess model generalization ability. In regular statistical models, the learning coefficient is given by d/2, where d is the dimension of the parameter space. More generally, it is defined as the absolute value of the pole order of a zeta function derived from the Kullback-Leibler divergence and the prior distribution. However, except for specific cases such as reduced-rank regression, the learning coefficient cannot be derived in a closed form. Watanabe proposed a numerical method to estimate the learning coefficient, which Imai further refined to enhance its convergence properties. These methods utilize the asymptotic behavior of WBIC and have been shown to be statistically consistent as the sample size grows. In this paper, we propose a novel numerical estimation method that fundamentally differs from previous approaches and leverages a new quantity, "Empirical Loss," which was introduced by Watanabe. Through numerical experiments, we demonstrate that our proposed method exhibits both lower bias and lower variance compared to those of Watanabe and Imai. Additionally, we provide a theoretical analysis that elucidates why our method outperforms existing techniques and present empirical evidence that supports our findings.

翻译：泛化误差系数在分析信息准则（如广适用信息准则WAIC和广适用贝叶斯信息准则WBIC）的性能中起着关键作用，这些准则由渡边澄夫提出，用于评估模型的泛化能力。在正则统计模型中，泛化误差系数由d/2给出，其中d为参数空间的维度。更一般地，它被定义为由Kullback-Leibler散度和先验分布导出的zeta函数的极点阶数的绝对值。然而，除降秩回归等特定情况外，泛化误差系数无法以闭式形式导出。渡边提出了一种数值方法来估计泛化误差系数，今井进一步改进了该方法以增强其收敛性。这些方法利用WBIC的渐近行为，已被证明在样本量增长时具有统计一致性。本文提出了一种全新的数值估计方法，其与先前方法存在根本性差异，并利用了渡边引入的新量——“经验损失”。通过数值实验，我们证明所提方法相较于渡边和今井的方法具有更低的偏差和方差。此外，我们提供了理论分析以阐明本方法优于现有技术的原因，并给出了支持该结论的实验证据。