From Physics to Statistics: A Simple Route to Exponential Families via Maximum Entropy

2026-04-24

AI Summary

This work addresses the lack of intuition in traditional derivations of exponential family distributions, which often obscure their information-theoretic and physical foundations in pedagogical contexts. By leveraging the principle of maximum entropy and requiring only elementary notions of entropy, the paper presents a concise and self-contained derivation that avoids complex constrained optimization. The core contribution demonstrates that, under constraints fixing the expected values of sufficient statistics, exponential family distributions uniquely maximize relative entropy with respect to a general base measure, and Shannon entropy in the special case of a uniform base measure. This approach reveals the fundamental connection between maximum entropy and exponential families from minimal assumptions, substantially streamlining the didactic exposition and fostering deeper integration of statistical theory with physical reasoning.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Exponential families form the backbone of modern statistics and machine learning, but textbooks seldom derive them from first principles in an accessible way. Although minimal sufficiency and the principle of maximum entropy, originating in physics, provide core motivation, they are often presented as technical and requiring advanced prerequisites. Here, a short, self-contained derivation of exponential families based on maximum entropy is presented that is straightforward to carry out, requires only a modest background in information entropy, and avoids technicalities like constrained optimisation. Two propositions are demonstrated in this fashion: i) exponential families with a general base maximise information entropy with respect to that base subject to fixed expectations of canonical statistics, and ii) exponential families with a uniform base maximise standard information entropy under the same constraints. Maximum entropy therefore provides a principled foundation for exponential families with minimal prerequisites, highlighting the value of teaching entropy in statistics courses.

Problem

• exponential families
• maximum entropy
• statistics education
• information entropy
• first principles

Innovation

• exponential families
• maximum entropy
• information entropy
• statistical derivation
• canonical statistics