[Paper Review] Natural selection. V. How to read the fundamental equations of evolutionary change in terms of information theory

Frank, Steven A.|arXiv (Cornell University)|Nov 16, 2012

Evolutionary Game Theory and Cooperation118 citations

TL;DR

This paper establishes a rigorous connection between natural selection and information theory by showing that the fundamental equations of evolutionary change—particularly Fisher's fundamental theorem—can be interpreted as measures of information accumulation. It demonstrates that the change in mean fitness is equivalent to the Jeffreys divergence, a symmetric information-theoretic measure, providing a deeper, more intuitive understanding of selection as the acquisition of environmental information.

ABSTRACT

The equations of evolutionary change by natural selection are commonly expressed in statistical terms. Fisher's fundamental theorem emphasizes the variance in fitness. Quantitative genetics expresses selection with covariances and regressions. Population genetic equations depend on genetic variances. How can we read those statistical expressions with respect to the meaning of natural selection? One possibility is to relate the statistical expressions to the amount of information that populations accumulate by selection. However, the connection between selection and information theory has never been compelling. Here, I show the correct relations between statistical expressions for selection and information theory expressions for selection. Those relations link selection to the fundamental concepts of entropy and information in the theories of physics, statistics, and communication. We can now read the equations of selection in terms of their natural meaning. Selection causes populations to accumulate information about the environment.

Motivation & Objective

• To resolve the long-standing disconnect between statistical expressions in evolutionary theory and their intuitive meaning by reinterpreting them through information theory.
• To show that the variance in fitness and other statistical measures in selection equations are equivalent to information-theoretic quantities such as Kullback-Leibler and Jeffreys divergences.
• To establish the Jeffreys divergence as the correct, general measure of information accumulated by natural selection, independent of small-change approximations.
• To clarify the distinction between selection (partial change) and total evolutionary dynamics by grounding information in meaningful parametric and nonparametric scales.
• To reframe Fisher information as the sensitivity of information gain with respect to changes in population parameters, linking it to statistical inference and information geometry.

Proposed method

• Reformulates classical selection equations (e.g., covariance between fitness and trait value) using information-theoretic measures, particularly the Jeffreys divergence.
• Demonstrates that the change in mean log-fitness equals the Jeffreys divergence between the initial and selected population distributions.
• Uses the Kullback-Leibler divergence and its symmetric form (Jeffreys divergence) to express information gain in selection, linking to entropy and relative entropy concepts.
• Applies Fisher information as the limiting form of the Jeffreys divergence under small changes, showing its role in measuring sensitivity of information to parameter shifts.
• Derives parametric and nonparametric expressions for selection, showing that information accumulation is universally measurable regardless of distributional assumptions.
• Establishes a conceptual framework where the scale of encoded information (log-likelihood) is linked to the scale of biological meaning (e.g., trait means or distributions).

Experimental results

Research questions

• RQ1 How can the standard statistical expressions of natural selection (e.g., variance in fitness) be reinterpreted in terms of information theory?
• RQ2 What is the correct information-theoretic measure that quantifies the amount of information a population accumulates through natural selection?
• RQ3 How does the Jeffreys divergence relate to Fisher’s fundamental theorem of natural selection and the concept of information gain?
• RQ4 Why is Fisher information the limiting form of the Jeffreys divergence, and what does this imply about the sensitivity of information to parameter changes?
• RQ5 How can the distinction between selection (partial change) and total evolutionary dynamics be clarified using information-theoretic frameworks?

Key findings

• The change in mean log-fitness due to selection is exactly equal to the Jeffreys divergence between the initial and selected population distributions.
• Fisher’s fundamental theorem—that the rate of increase in mean fitness equals the genetic variance in fitness—is equivalent to the statement that selection accumulates information measured by the Jeffreys divergence.
• The Jeffreys divergence provides a general, non-asymptotic measure of information gain by selection, valid even for large changes, unlike Fisher information which is limited to small changes.
• Fisher information emerges as the limiting form of the Jeffreys divergence when changes in population parameters are small, linking selection to statistical inference and the curvature of the log-likelihood.
• The information-theoretic interpretation clarifies that selection is not about time or dynamics per se, but about the accumulation of meaningful information about the environment through changes in gene frequencies.
• The framework unifies parametric and nonparametric descriptions of selection by showing that information gain is measurable regardless of whether changes are summarized by parameters or full distributions.