---
tag: F64Z9541DV
kind: concept
name: Information geometry
msc: ["53B12", "62B11"]
sources:
  - url: https://en.wikipedia.org/wiki/Information_geometry
same_as:
  wikidata: Q3798604
  wikipedia: Information geometry
---

## Summary

Information geometry studies families of [[probability distributions|0DVDCMA2DK]] as manifolds, using the tools of differential geometry.
The central structure is the Fisher information metric, a [[Riemannian metric|Q61VFHCH2J]] on the parameter space, together with a family of [[affine connections|JX65KSVHSJ]] called the $\alpha$-connections.
On finite sample spaces, Chentsov's theorem shows that the Fisher metric is, up to a constant factor, the only metric invariant under sufficient statistics.
The field was developed largely by Shun-ichi Amari and has applications in statistics, machine learning and information theory.

## Explanations

### L1 human

A statistical model is a family of distributions described by a few parameters, like the mean and standard deviation of a normal distribution.
Each choice of parameters is a point, so the model is a space of points.
Information geometry measures the distance between two nearby points by how easily data can tell the two distributions apart.
For normal distributions, two distributions with means $0$ and $1$ are far apart when the standard deviation is small and close when it is large.
With this distance the family of normal distributions becomes a copy of the hyperbolic plane.