Information geometry

Source: Wikipedia, the free encyclopedia.
The set of all normal distributions forms a statistical manifold with hyperbolic geometry.

Information geometry is an interdisciplinary field that applies the techniques of

probability distributions
.

Introduction

Historically, information geometry can be traced back to the work of

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.

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 [8]
  • Time series and linear systems
  • Filtering problem[9]
  • Quantum systems[10]
  • Neural networks
  • Machine learning
  • Statistical mechanics
  • Biology
  • Statistics [11] [12]
  • Mathematical finance [13]

See also

References

  1. ^ Nielsen, Frank (2022). "The Many Faces of Information Geometry" (PDF). Notices of the AMS. 69 (1). American Mathematical Society: 36-45.
  2. S2CID 117034671
    .
  3. S2CID 16759683
    .
  4. doi:10.1002/ecja.4400660602
    .
  5. ISBN 0-8218-0531-2
    .
  6. ISBN 978-3-319-56477-7
    .
  7. ^ Nielsen, Frank (2018). "An Elementary Introduction to Information Geometry". Entropy. 22 (10).
  8. ISBN 0-471-82668-5
    .
  9. doi:10.1109/9.661075
    .
  10. S2CID 15292186
    .
  11. ISBN 0-387-96056-2
    .
  12. ISBN 0-412-39860-5
    .
  13. ISBN 0-521-65116-6
    .

External links
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