Information geometry explained

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.

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.^{[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.

• Ronald Fisher
• Harald Cramér
• Calyampudi Radhakrishna Rao
  • Harold Jeffreys Solomon Kullback
• Jean-Louis Koszul
• Richard Leibler
• Claude Shannon
• Imre Csiszár
• Nikolai Chentsov (also written as N. N. Čencov)
• Bradley Efron
• Shun'ichi Amari
• Ole Barndorff-Nielsen
• Frank Nielsen
• Damiano Brigo
• A. W. F. Edwards
• Grant Hillier
• Kees Jan van Garderen

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^[11]
• Machine learning
• Statistical mechanics
• Biology
• Statistics ^{[12] [13]}
• Mathematical finance ^[14]

See also

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

External links

• https://www.springer.com/mathematics/geometry/journal/41884 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

Notes and References

1. Frank . Nielsen . The Many Faces of Information Geometry. Notices of the AMS. American Mathematical Society . 2022. 69 . 1. 36-45.
2. Rao . C. R. . 1945 . Information and Accuracy Attainable in the Estimation of Statistical Parameters . Bulletin of the Calcutta Mathematical Society . 37 . 81–91 . Reprinted in Book: Breakthroughs in Statistics . Springer . 1992 . 235–247 . 10.1007/978-1-4612-0919-5_16 . 117034671 .
3. Book: Nielsen, F. . 2013 . 1301.3578 . Cramér-Rao Lower Bound and Information Geometry . Connected at Infinity II: On the Work of Indian Mathematicians . Texts and Readings in Mathematics . R. . Bhatia . C. S. . Rajan . Special Volume of Texts and Readings in Mathematics (TRIM) . 18–37 . Hindustan Book Agency . 10.1007/978-93-86279-56-9_2 . 978-93-80250-51-9 . 16759683 .
4. Shun'ichi . Amari . A foundation of information geometry . Electronics and Communications in Japan . 1983 . 66 . 6 . 1–10 . 10.1002/ecja.4400660602 .
5. Book: Shun'ichi . Amari . Hiroshi . Nagaoka . Methods of Information Geometry . Translations of Mathematical Monographs . 191 . American Mathematical Society . 2000 . 0-8218-0531-2 .
6. Book: Nihat . Ay . Jürgen . Jost . Jürgen Jost . Hông Vân . Lê . Lorenz . Schwachhöfer . Information Geometry . 64 . Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer . 2017 . 978-3-319-56477-7 .
7. Frank . Nielsen . Entropy . An Elementary Introduction to Information Geometry . 2018 . 22 . 10 .
8. Book: R. E. . Kass . P. W. . Vos . 1997 . Geometrical Foundations of Asymptotic Inference . Series in Probability and Statistics . Wiley . 0-471-82668-5 .
9. Brigo . Damiano . Hanzon . Bernard . LeGland . Francois . 1998 . A differential geometric approach to nonlinear filtering: the projection filter . IEEE Transactions on Automatic Control . 43 . 2 . 247–252 . 10.1109/9.661075 . Damiano Brigo .
10. van Handel . Ramon . Mabuchi . Hideo . 2005 . Quantum projection filter for a highly nonlinear model in cavity QED . Journal of Optics B: Quantum and Semiclassical Optics. 7 . 10 . S226–S236 . 10.1088/1464-4266/7/10/005 . quant-ph/0503222 . 2005JOptB...7S.226V . 15292186 .
11. Zlochin . Mark . Baram . Yoram . 2001 . Manifold Stochastic Dynamics for Bayesian Learning . Neural Computation. 13 . 11 . 2549–2572 . 10.1162/089976601753196021 .
12. Book: Amari, Shun'ichi . 1985 . Differential-Geometrical Methods in Statistics . Lecture Notes in Statistics . Springer-Verlag . Berlin . 0-387-96056-2 .
13. Book: M. . Murray . J. . Rice . 1993 . Differential Geometry and Statistics . Monographs on Statistics and Applied Probability . 48 . . 0-412-39860-5 .
14. Book: Paul . Marriott . Mark . Salmon . Applications of Differential Geometry to Econometrics . Cambridge University Press . 2000 . 0-521-65116-6 .