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Technique in statistics From Wikipedia, the free encyclopedia
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. [1] It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions.
This article may need to be rewritten to comply with Wikipedia's quality standards, as This should be edited to include some statistical background.. (April 2019) |
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.[2][3] The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.[4]
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.
The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book, Methods of Information Geometry,[5] and the more recent book by Nihat Ay and others.[6] A gentle introduction is given in the survey by Frank Nielsen.[7] In 2018, the journal Information Geometry was released, which is devoted to the field.
This section may need to be rewritten to comply with Wikipedia's quality standards, as Does this list make sense in a general article?. (May 2013) |
The history of information geometry is associated with the discoveries of at least the following people, and many others.
This section may need to be rewritten to comply with Wikipedia's quality standards, as References should be provided for these.. (April 2019) |
As an interdisciplinary field, information geometry has been used in various applications.
Here an incomplete list:
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