Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally. At the same time, widely used criteria such as WAIC and WBIC appear disconnected from underlying singular geometry. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions. A universal covariance identity links derivatives of tempered expectations to posterior fluctuations, placing WAIC, WBIC, and singular fluctuation within a unified response framework. Within this framework, classical quantities from singular learning theory acquire natural thermodynamic interpretations: RLCT governs the leading free-energy slope, singular fluctuation corresponds to curvature of the tempered free energy, and WAIC measures predictive fluctuation. We formalize an observable algebra that quotients out non-identifiable directions, allowing structurally meaningful order parameters to be constructed in singular models. Across canonical singular examples-including symmetric Gaussian mixtures, reduced-rank regression, and overparameterized neural networks-we empirically demonstrate phase-transition-like behavior under tempering. Order parameters collapse, susceptibilities peak, and complexity measures align with structural reorganization in posterior geometry. Our results suggest that thermodynamic response theory provides a natural organizing framework for interpreting complexity, predictive variability, and structural reorganization in singular Bayesian learning.

翻译：奇异统计模型——包括混合模型、矩阵分解和神经网络——由于参数不可识别性和退化的费希尔几何结构而违反正则渐近性。尽管奇异学习理论通过实对数典范阈值和奇异涨落等不变量刻画了边缘似然行为，但这些量在操作层面仍难以解释。同时，广泛使用的WAIC和WBIC准则似乎与底层奇异几何结构脱节。本文证明后验回火会诱导产生单参数的后验分布形变，其关联可观测量生成热力学响应函数的层级结构。一个普适的协方差恒等式将回火期望的导数与后验涨落联系起来，使WAIC、WBIC和奇异涨落统一于响应框架之中。在此框架内，奇异学习理论的经典量获得了自然的热力学诠释：RLCT主导自由能的主导斜率，奇异涨落对应回火自由能的曲率，而WAIC度量预测涨落。我们形式化了一个可观测量代数，该代数商除了不可识别方向，使得在奇异模型中能够构建具有结构意义的序参数。在典型奇异示例（包括对称高斯混合模型、降秩回归和过参数化神经网络）中，我们通过实验证明了回火作用下类似相变的行为：序参数坍缩、磁化率峰值出现，且复杂度度量与后验几何的结构重组相一致。我们的结果表明，热力学响应理论为解释奇异贝叶斯学习中的复杂度、预测变异性和结构重组提供了自然的组织框架。