A Thermodynamic Structure of Asymptotic Inference

TL;DR

This paper introduces a thermodynamic framework for asymptotic inference, linking information theory and thermodynamics to derive bounds, efficiencies, and fundamental limits in statistical estimation.

Contribution

It develops a novel thermodynamic structure for asymptotic inference, revealing new bounds, efficiency limits, and connections to physical principles like Carnot cycles.

Findings

Derived a first-law-like balance equation for inference.

Established a reversed second law inequality for estimation.

Identified a lower bound on entropy analogous to a third law.

Abstract

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.