The Fisher information depends on the parametrization of the problem. If θ and η are two scalar parametrizations of an estimation problem, and θ is a continuously differentiable function of η, thenwhere and are the Fisher information measures of η and θ, respectively.^{[12]}In the vector case, suppose and arek-vectors which parametrize an estimation problem, and suppose that is a continuously differentiable function of , then,^{[13]}where the (i,j)th element of thek×kJacobian matrix is defined byand where is the matrix transpose of .In information geometry, this is seen as a change of coordinates on a Riemannian manifold, and the intrinsic properties of curvature are unchanged under different parametrization. In general, the Fisher information matrix provides a Riemannian metric (more precisely, the Fisher-Rao metric) for the manifold of thermodynamic states, and can be used as aninformation-geometric complexity measurefor a classification of phase transitions, e.g., the scalar curvature of the thermodynamic metric tensor diverges at (and only at) a phase transition point.^{[14]}In the thermodynamic context, the Fisher information matrix is directly related to the rate of change in the corresponding order parameters[emphasis mine].^{[15]}In particular, such relations identify second-order phase transitions via divergences of individual elements of the Fisher information matrix.

**I wonder how this could be extended to the dispersion of mutations through configuration space.**

Some possible sources for further inquiry:

- Information Geometry by Arwini and Dodson
- Information, Physics and Computation by Mezard and Montanari

Also

**Information Geometry: From Black Holes to Condensed Matter Systems**, an editorial by Tapobrata Sarkar, Hernando Quevedo, and Rong-Gen Cai.

And for black hole thermodynamics look at

**Information geometries for black hole physics**by

**, as well as**

__Narit Pidokrajt__**Information Geometric Approach to Black Hole Thermodynamics**by the same author.

The figure below is from Frederic Barbaresco's "Eidetic Reduction of Information Geometry" in

**Geometric Theory of Information**by Frank Nielsen.