The Price equation program: simple invariances unify population dynamics, thermodynamics, probability, information and inference

Abstract

The fundamental equations of various disciplines often seem to share the same basic structure. Natural selection increases information in the same way that Bayesian updating increases information. Thermodynamics and the forms of common probability distributions express maximum increase in entropy, which appears mathematically as loss of information. Physical mechanics follows paths of change that maximize Fisher information. The information expressions typically have analogous interpretations as the Newtonian balance between force and acceleration, representing a partition between direct causes of change and opposing changes in the frame of reference. This web of vague analogies hints at a deeper common mathematical structure. I suggest that the Price equation expresses that underlying universal structure. The abstract Price equation describes dynamics as the change between two sets. One component of dynamics expresses the change in the frequency of things, holding constant the values associated with things. The other component of dynamics expresses the change in the values of things, holding constant the frequency of things. The separation of frequency from value generalizes Shannon's separation of the frequency of symbols from the meaning of symbols in information theory. The Price equation's generalized separation of frequency and value reveals a few simple invariances that define universal geometric aspects of change. For example, the conservation of total frequency, although a trivial invariance by itself, creates a powerful constraint on the geometry of change. That constraint plus a few others seem to explain the common structural forms of the equations in different disciplines. From that abstract perspective, interpretations such as selection, information, entropy, force, acceleration, and physical work arise from the same underlying geometry expressed by the Price equation.

arXiv commentVersion 3: added figure illustrating geometry; added table of symbols and two tables summarizing mathematical relations; this version accepted for publication in Entropy

References in corpus (9)

• The common patterns of nature
• Natural selection. IV. The Price equation
• Natural selection maximizes Fisher information
• Dynamics of Bayesian Updating with Dependent Data and Misspecified Models
• Universal expressions of population change by the Price equation: natural selection, information, and maximum entropy production
• Common probability patterns arise from simple invariances
• d'Alembert's direct and inertial forces acting on populations: the Price equation and the fundamental theorem of natural selection
• How to read probability distributions as statements about process
• Invariant death

Cited by in corpus (5)

• Geometric thermodynamics for the Fokker-Planck equation: Stochastic thermodynamic links between information geometry and optimal transport
• Simple unity among the fundamental equations of science
• The fundamental equations of change in statistical ensembles and biological populations
• Thermodynamics of Evolution and the Origin of Life
• The Price equation reveals a universal force-metric-bias law of algorithmic learning and natural selection