Legendre transformation and information geometry for the maximum entropy theory of ecology

TL;DR

This paper explores the mathematical structure of the maximum entropy theory of ecology (METE) using information geometry, deriving analytical expressions for macrostate entropy and related geometric metrics.

Contribution

It introduces a novel analytical calculation of macrostate entropy via Legendre transformation, linking METE to information geometry.

Findings

Analytical macrostate entropy derived from Legendre transformation.

Calculation of metric terms and covariance matrix in METE.

Enhanced understanding of METE's geometric structure.

Abstract

Here I investigate some mathematical aspects of the maximum entropy theory of ecology (METE). In particular I address the geometrical structure of METE endowed by information geometry. As novel results, the macrostate entropy is calculated analytically by the Legendre transformation of the log-normalizer in METE. This result allows for the calculation of the metric terms in the information geometry arising from METE and, by consequence, the covariance matrix between METE variables.