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：本文发展了一个用于渐近推断的热力学框架，其中样本量与参数方差定义了一个状态空间。在此描述中，香农信息扮演熵的角色，而一个积分因子将其变化组织成一个第一定律类型的平衡方程。该框架支持一个类似于反向第二定律的循环不等式，该不等式是针对均值估计推导得出的。一个非平凡的三定律类型结果以表示噪声设定的熵下界形式出现。最优推断路径、信息增益的全局界限以及一种自然的类卡诺信息效率均由此结构导出，其效率从根本上受到噪声底限的限制。最后，高斯极限情形下的德布鲁因恒等式与I-MMSE关系表现为同一基础热力学结构的坐标投影。该框架表明，系综物理学与推断物理学构成了在统一热力学描述中沿相反方向演化的影子过程。