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Information geometry
Field that applies the techniques of differential geometry to study probability theory and statistics.

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.

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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.23 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, 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 in information theory and machine learning. 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 descent6 and natural gradient descent7).

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 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 11
  • Time series and linear systems
  • Filtering problem12
  • Quantum systems13
  • Neural networks14
  • Machine learning
  • Statistical mechanics
  • Biology
  • Statistics 15 16
  • Mathematical finance 17

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. https://www.ams.org/journals/notices/202201/rnoti-p36.pdf ↩

  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 Breakthroughs in Statistics. Springer. 1992. pp. 235–247. doi:10.1007/978-1-4612-0919-5_16. S2CID 117034671. /wiki/Doi_(identifier) ↩

  3. Nielsen, F. (2013). "Cramér-Rao Lower Bound and Information Geometry". In Bhatia, R.; Rajan, C. S. (eds.). Connected at Infinity II: On the Work of Indian Mathematicians. Texts and Readings in Mathematics. Vol. Special Volume of Texts and Readings in Mathematics (TRIM). Hindustan Book Agency. pp. 18–37. arXiv:1301.3578. doi:10.1007/978-93-86279-56-9_2. ISBN 978-93-80250-51-9. S2CID 16759683. 978-93-80250-51-9 ↩

  4. Amari, Shun'ichi (1983). "A foundation of information geometry". Electronics and Communications in Japan. 66 (6): 1–10. doi:10.1002/ecja.4400660602. https://onlinelibrary.wiley.com/doi/abs/10.1002/ecja.4400660602 ↩

  5. Bauer, Martin; Le Brigant, Alice; Lu, Yuxiu; Maor, Cy (2024-02-10). "The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections". Calculus of Variations and Partial Differential Equations. 63 (2): 56. arXiv:2306.14533. doi:10.1007/s00526-024-02660-5. ISSN 1432-0835. https://link.springer.com/article/10.1007/s00526-024-02660-5 ↩

  6. Raskutti, Garvesh; Mukherjee, Sayan (March 2015). "The Information Geometry of Mirror Descent". IEEE Transactions on Information Theory. 61 (3): 1451–1457. arXiv:1310.7780. doi:10.1109/TIT.2015.2388583. ISSN 0018-9448. https://ieeexplore.ieee.org/document/7004065 ↩

  7. Abdulkadirov, Ruslan; Lyakhov, Pavel; Nagornov, Nikolay (January 2022). "Accelerating Extreme Search of Multidimensional Functions Based on Natural Gradient Descent with Dirichlet Distributions". Mathematics. 10 (19): 3556. doi:10.3390/math10193556. ISSN 2227-7390. https://doi.org/10.3390%2Fmath10193556 ↩

  8. Amari, Shun'ichi; Nagaoka, Hiroshi (2000). Methods of Information Geometry. Translations of Mathematical Monographs. Vol. 191. American Mathematical Society. ISBN 0-8218-0531-2. 0-8218-0531-2 ↩

  9. Ay, Nihat; Jost, Jürgen; Lê, Hông Vân; Schwachhöfer, Lorenz (2017). Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 64. Springer. ISBN 978-3-319-56477-7. 978-3-319-56477-7 ↩

  10. Nielsen, Frank (2018). "An Elementary Introduction to Information Geometry". Entropy. 22 (10). https://www.mdpi.com/1099-4300/22/10/1100 ↩

  11. Kass, R. E.; Vos, P. W. (1997). Geometrical Foundations of Asymptotic Inference. Series in Probability and Statistics. Wiley. ISBN 0-471-82668-5. 0-471-82668-5 ↩

  12. Brigo, Damiano; Hanzon, Bernard; LeGland, Francois (1998). "A differential geometric approach to nonlinear filtering: the projection filter" (PDF). IEEE Transactions on Automatic Control. 43 (2): 247–252. doi:10.1109/9.661075. /wiki/Damiano_Brigo ↩

  13. 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. arXiv:quant-ph/0503222. Bibcode:2005JOptB...7S.226V. doi:10.1088/1464-4266/7/10/005. S2CID 15292186. /wiki/ArXiv_(identifier) ↩

  14. Zlochin, Mark; Baram, Yoram (2001). "Manifold Stochastic Dynamics for Bayesian Learning". Neural Computation. 13 (11): 2549–2572. doi:10.1162/089976601753196021. PMID 11674851. https://direct.mit.edu/neco/article-abstract/13/11/2549/6467/Manifold-Stochastic-Dynamics-for-Bayesian-Learning?redirectedFrom=fulltext ↩

  15. Amari, Shun'ichi (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Berlin: Springer-Verlag. ISBN 0-387-96056-2. 0-387-96056-2 ↩

  16. Murray, M.; Rice, J. (1993). Differential Geometry and Statistics. Monographs on Statistics and Applied Probability. Vol. 48. Chapman and Hall. ISBN 0-412-39860-5. 0-412-39860-5 ↩

  17. Marriott, Paul; Salmon, Mark, eds. (2000). Applications of Differential Geometry to Econometrics. Cambridge University Press. ISBN 0-521-65116-6. 0-521-65116-6 ↩