Singular learning theory (SLT) \citep{watanabe2009algebraic,watanabe2018mathematical} provides a rigorous asymptotic framework for Bayesian models with non-identifiable parameterizations, yet the statistical meaning of its second-order invariant, the \emph{singular fluctuation}, has remained unclear. In this work, we show that singular fluctuation admits a precise and natural interpretation as a \emph{specific heat}: the second derivative of the Bayesian free energy with respect to temperature. Equivalently, it measures the posterior variance of the log-likelihood observable under the tempered Gibbs posterior. We further introduce a collection of related thermodynamic quantities, including entropy flow, prior susceptibility, and cross-susceptibility, that together provide a detailed geometric diagnosis of singular posterior structure. Through extensive numerical experiments spanning discrete symmetries, boundary singularities, continuous gauge freedoms, and piecewise (ReLU) models, we demonstrate that these thermodynamic signatures cleanly distinguish singularity types, exhibit stable finite-sample behavior, and reveal phase-transition--like phenomena as temperature varies. We also show empirically that the widely used WAIC estimator \citep{watanabe2010asymptotic, watanabe2013widely} is exactly twice the thermodynamic specific heat at unit temperature, clarifying its robustness in singular models.Our results establish a concrete bridge between singular learning theory and statistical mechanics, providing both theoretical insight and practical diagnostics for modern Bayesian models.

翻译：奇异学习理论（SLT）\citep{watanabe2009algebraic,watanabe2018mathematical}为具有不可识别参数化的贝叶斯模型提供了一个严格的渐近框架，然而其二阶不变量——\emph{奇异涨落}——的统计意义一直不明确。本文证明，奇异涨落可被精确且自然地解释为一种\emph{比热}：即贝叶斯自由能对温度的二次导数。等价地，它度量了在调温吉布斯后验下对数似然可观测量后验方差。我们进一步引入了一系列相关的热力学量，包括熵流、先验磁化率与交叉磁化率，它们共同提供了对奇异后验结构的详细几何诊断。通过涵盖离散对称性、边界奇异性、连续规范自由度及分段（ReLU）模型的广泛数值实验，我们证明这些热力学特征能清晰区分奇异性类型，展现出稳定的有限样本行为，并在温度变化时揭示出类似相变的现象。我们还通过实证表明，广泛使用的WAIC估计量\citep{watanabe2010asymptotic, watanabe2013widely}恰好等于单位温度下热力学比热的两倍，从而阐明了其在奇异模型中的稳健性。我们的研究结果在奇异学习理论与统计力学之间建立了具体的桥梁，为现代贝叶斯模型提供了理论洞见与实践诊断工具。