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Entropy and Diversity: The Axiomatic Approach

Tom Leinster

TL;DR

Entropy and Diversity develops an axiomatic framework to quantify biodiversity by tying diversity to information-theoretic entropy and its deformations. It derives Shannon entropy from functional equations, then extends to one-parameter deformations (the $q$-logarithms and Rényi entropies) leading to Hill numbers as effective diversity measures, and introduces similarity- and cross-diversity concepts. The work further links these ideas to magnitude in enriched categories, information geometry, and measure-theoretic generalizations, creating a unified approach that spans ecology, probability, and pure mathematics. The resulting theorems—such as the chain-rule characterization of entropy and the unique determination of Hill numbers by axioms—provide rigorous, interpretable tools for comparing, aggregating, and understanding diversity across contexts, with broad implications for metacommunities and microbial ecology.

Abstract

This book brings new mathematical rigour to the ongoing vigorous debate on how to quantify biological diversity. The question "what is diversity?" has surprising mathematical depth, and breadth too: this book involves parts of mathematics ranging from information theory, functional equations and probability theory to category theory, geometric measure theory and number theory. It applies the power of the axiomatic method to a biological problem of pressing concern, but the new concepts and theorems are also motivated from a purely mathematical perspective. The main narrative thread requires no more than an undergraduate course in analysis. No familiarity with entropy or diversity is assumed.

Entropy and Diversity: The Axiomatic Approach

TL;DR

Entropy and Diversity develops an axiomatic framework to quantify biodiversity by tying diversity to information-theoretic entropy and its deformations. It derives Shannon entropy from functional equations, then extends to one-parameter deformations (the -logarithms and Rényi entropies) leading to Hill numbers as effective diversity measures, and introduces similarity- and cross-diversity concepts. The work further links these ideas to magnitude in enriched categories, information geometry, and measure-theoretic generalizations, creating a unified approach that spans ecology, probability, and pure mathematics. The resulting theorems—such as the chain-rule characterization of entropy and the unique determination of Hill numbers by axioms—provide rigorous, interpretable tools for comparing, aggregating, and understanding diversity across contexts, with broad implications for metacommunities and microbial ecology.

Abstract

This book brings new mathematical rigour to the ongoing vigorous debate on how to quantify biological diversity. The question "what is diversity?" has surprising mathematical depth, and breadth too: this book involves parts of mathematics ranging from information theory, functional equations and probability theory to category theory, geometric measure theory and number theory. It applies the power of the axiomatic method to a biological problem of pressing concern, but the new concepts and theorems are also motivated from a purely mathematical perspective. The main narrative thread requires no more than an undergraduate course in analysis. No familiarity with entropy or diversity is assumed.

Paper Structure

This paper contains 76 sections, 208 theorems, 1290 equations, 40 figures, 7 tables.

Table of Contents

  1. Fundamental functional equations
  2. Cauchy's equation
  3. Logarithmic sequences
  4. The $q$-logarithm
  5. Shannon entropy
  6. Probability distributions on finite sets
  7. Definition and properties of Shannon entropy
  8. Entropy in terms of coding
  9. Entropy in terms of diversity
  10. The chain rule characterizes entropy
  11. Relative entropy
  12. Definition and properties of relative entropy
  13. Relative entropy in terms of coding
  14. Relative entropy in terms of diversity
  15. Relative entropy in measure theory, geometry and statistics
  16. ...and 61 more sections

Key Result

Proposition 1.1.1

Every differentiable additive function $\mathbb{R} \to \mathbb{R}$ is linear.

Figures (40)

  • Figure 2.1: Composition of probability distributions.
  • Figure 2.2: The composite distribution of Example \ref{['eg:comp-coin']}.
  • Figure 2.3: The composite distribution for French symbols (Example \ref{['eg:comp-french']}).
  • Figure 2.4: Four probability distributions on $\{1, 2, 3, 4\}$, and their entropies to base $2$.
  • Figure 2.5: The entropy of the French language (Example \ref{['eg:ent-french']}).
  • ...and 35 more figures

Theorems & Definitions (604)

  • Proposition 1.1.1
  • Proof 1
  • Lemma 1.1.2
  • Proof 2
  • Remark 1.1.3
  • Proposition 1.1.4
  • Proof 3
  • Proposition 1.1.5
  • Proof 4
  • Remark 1.1.6
  • ...and 594 more