Statistical learning models such as multilayer neural networks and mixed distributions are widely used, and understanding the accuracy of these models is crucial for their use. Recent advances have clarified theoretical learning accuracy in Bayesian inference, where metrics such as generalization loss and free energy are used to measure the accuracy of predictive distributions. It has become clear that the asymptotic behavior of these metrics is determined by a rational number specific to each statistical model, known as the learning coefficient (real log canonical threshold). The problem of determining the learning coefficient is known to be reducible to the problem of finding the normal crossing of Kullback-Leibler divergence in relation to algebraic geometry. In this context, it is crucial to perform appropriate coordinate transformations and blow-ups. This paper attempts to derive appropriate variable transformations and blow-ups from the properties of the log-likelihood ratio function. That is, instead of dealing with the Kullback-Leibler information itself, it uses the properties of the log-likelihood ratio function before taking the expectation to calculate the real log canonical threshold. This approach has not been considered in previous research. Using these variable transformations and blow-ups, this paper provides the exact values of the learning coefficients and their calculation methods for statistical models that meet simple conditions next to the regular conditions (referred to as semi-regular models), and as specific examples, provides the learning coefficients for semi-regular models with two parameters and for those models where the random variables take a finite number of values.

翻译：多层神经网络和混合分布等统计学习模型被广泛应用，理解这些模型的精度对其使用至关重要。近期研究进展阐明了贝叶斯推断中的理论学习精度，其中泛化损失和自由能等度量被用于衡量预测分布的准确性。现已明确，这些度量的渐近行为由每个统计模型特有的有理数决定，该数被称为学习系数（实对数典范阈值）。确定学习系数的问题已知可归结为寻找Kullback-Leibler散度关于代数几何的正规交叉问题。在此背景下，执行适当的坐标变换和爆破变换至关重要。本文尝试从对数似然比函数的性质推导出合适的变量变换与爆破变换。即不直接处理Kullback-Leibler信息本身，而是利用取期望前的对数似然比函数性质来计算实对数典范阈值。该方法在以往研究中尚未被考虑。通过运用这些变量变换与爆破变换，本文针对满足正则条件旁侧简单条件的统计模型（称为半正则模型），给出了学习系数的精确值及其计算方法，并以双参数半正则模型和随机变量取有限值情形的半正则模型作为具体示例，提供了相应的学习系数。