• Source: Information geometry

Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions.




Introduction



Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.
Classically, information geometry considered a parametrized statistical model as a Riemannian manifold. For such models, there is a natural choice of Riemannian metric, known as the Fisher information metric. In the special case that the statistical model is an exponential 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 flat affine connections, as well as a canonical Bregman 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 from information theory, affine differential geometry, convex analysis and many other fields. One of the most perspective information geometry approaches find applications in machine learning. For example, the developing of information-geometric optimization methods (mirror descent and natural gradient descent).
The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book, Methods of Information Geometry, and the more recent book by Nihat Ay and others. A gentle introduction is given in the survey by Frank Nielsen. In 2018, the journal Information Geometry was released, which is devoted to the field.




Contributors



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




Applications



As an interdisciplinary field, information geometry has been used in various applications.
Here an incomplete list:

Statistical inference
Time series and linear systems
Filtering problem
Quantum systems
Neural networks
Machine learning
Statistical mechanics
Biology
Statistics
Mathematical finance




See also


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




References






External links


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

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• Jörg-Rüdiger Sack
• Calyampudi Radhakrishna Rao
• Persegi panjang
• Ilmu komputer teoretis
• Fraktal
• Segitiga sama kaki
• Persegi
• Layang-layang (geometri)
• Keterkaitan kuantum
• Joseph O'Rourke
• Information geometry
• Geometry
• Manifold hypothesis
• Fisher information metric
• Kullback–Leibler divergence
• Information
• Fisher information
• Differential geometry
• Central tendency
• Triangle