Information geometry is an interdisciplinary field that applies the techniques ofdifferential geometry to studyprobability theory andstatistics.^[1] It studiesstatistical manifolds, which areRiemannian manifolds whose points correspond toprobability distributions.

This articlemay need to be rewritten to comply with Wikipedia'squality standards, as This should be edited to include some statistical background.. Relevant discussion may be found on thetalk page.You can help. The talk page may contain suggestions.(April 2019)

Historically, information geometry can be traced back to the work ofC. R. Rao, who was the first to treat theFisher matrix as aRiemannian metric.^[2][3] The modern theory is largely due toShun'ichi Amari, whose work has been greatly influential on the development of the field.^[4]

Classically, information geometry considered a parametrizedstatistical model as aRiemannian, conjugate connection, statistical, and dually flat manifolds. Unlike usual smooth manifolds with tensor metric and Levi-Civita connection, these take into account conjugate connection, torsion, and Amari-Chentsov metric.^[5] All presented above geometric structures find application ininformation theory andmachine learning. For such models, there is a natural choice of Riemannian metric, known as theFisher information metric. In the special case that the statistical model is anexponential family, it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function). In this case, the manifold naturally inherits two flataffine connections, as well as a canonicalBregman divergence. Historically, much of the work was devoted to studying the associated geometry of these examples. In the modern setting, information geometry applies to a much wider context, including non-exponential families,nonparametric statistics, and even abstract statistical manifolds not induced from a known statistical model. The results combine techniques frominformation theory,affine differential geometry,convex analysis and many other fields. One of the most perspective information geometry approaches find applications inmachine learning. For example, the developing of information-geometric optimization methods (mirror descent^[6] and natural gradient descent^[7]).

The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book,Methods of Information Geometry,^[8] and the more recent book by Nihat Ay and others.^[9] A gentle introduction is given in the survey by Frank Nielsen.^[10] In 2018, the journalInformation Geometry was released, which is devoted to the field.

This sectionmay need to be rewritten to comply with Wikipedia'squality standards, as Does this list make sense in a general article?. Relevant discussion may be found on thetalk page.You can help. The talk page may contain suggestions.(May 2013)

The history of information geometry is associated with the discoveries of at least the following people, and many others.

This sectionmay need to be rewritten to comply with Wikipedia'squality standards, as References should be provided for these.. Relevant discussion may be found on thetalk page.You can help. The talk page may contain suggestions.(April 2019)

As an interdisciplinary field, information geometry has been used in various applications.

Here an incomplete list:

• Statistical inference^[11]
• Time series and linear systems
• Filtering problem^[12]
• Quantum systems^[13]
• Neural networks^[14]
• Machine learning
• Statistical mechanics
• Biology
• Statistics^[15][16]
• Mathematical finance^[17]

• Ruppeiner geometry
• Kullback–Leibler divergence
• Stochastic geometry
• Stochastic differential geometry
• Projection filters

• [1] Information Geometry journal by Springer
• Information Geometry overview by Cosma Rohilla Shalizi, July 2010
• Information Geometry notes byJohn Baez, November 2012
• Information geometry for neural networks(pdf ), by Daniel Wagenaar

• Information geometry

• CS1: long volume value
• Articles with short description
• Short description is different from Wikidata
• Wikipedia articles needing rewrite from April 2019
• All articles needing rewrite
• Wikipedia articles needing rewrite from May 2013