A thermodynamic framework for asymptotic inference is developed in which sample size and parameter variance define a state space. Within this description, Shannon information plays the role of entropy, and an integrating factor organizes its variation into a first-law-type balance equation. The framework supports a cyclic inequality analogous to a reversed second law, derived for the estimation of the mean. A non-trivial third-law-type result emerges as a lower bound on entropy set by representation noise. Optimal inference paths, global bounds on information gain, and a natural Carnot-like information efficiency follow from this structure, with efficiency fundamentally limited by a noise floor. Finally, de Bruijn's identity and the I-MMSE relation in the Gaussian-limit case appear as coordinate projections of the same underlying thermodynamic structure. This framework suggests that ensemble physics and inferential physics constitute shadow processes evolving in opposite directions within a unified thermodynamic description.

翻译：我们发展了一种用于渐近推断的热力学框架，其中样本量和参数方差定义了状态空间。在此描述中，香农信息扮演熵的角色，而积分因子将其变化组织成类似热力学第一定律的平衡方程。该框架支持一个类似于逆向热力学第二定律的循环不等式，它是在均值估计的背景下推导出的。一个非平庸的类似热力学第三定律的结果作为由表示噪声设定的熵下界而出现。从这一结构中推导出最优推断路径、信息增益的全局界限以及自然的类似卡诺的信息效率，其效率从根本上受到噪声底限的限制。最后，在高斯极限情形下，德布罗因恒等式和I-MMSE关系表现为同一底层热力学结构的坐标投影。该框架表明，在统一的热力学描述中，系综物理学与推断物理学构成了沿相反方向演化的影过程。