Information geometry

{{Cleanup rewrite|This should be edited to include some statistical background.|article|date=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.{{cite journal |last=Rao |first=C. R. |year=1945 |title=Information and Accuracy Attainable in the Estimation of Statistical Parameters |journal=Bulletin of the Calcutta Mathematical Society |volume=37 |pages=81–91 }} Reprinted in {{cite book |title=Breakthroughs in Statistics |publisher=Springer |year=1992 |pages=235–247 |doi=10.1007/978-1-4612-0919-5_16 |s2cid=117034671 }}{{cite book |first=F. |last=Nielsen |year=2013 |arxiv=1301.3578 |chapter=Cramér-Rao Lower Bound and Information Geometry |title=Connected at Infinity II: On the Work of Indian Mathematicians |series=Texts and Readings in Mathematics |editor-first=R. |editor1-last=Bhatia |editor2-first=C. S. |editor2-last=Rajan |volume=Special Volume of Texts and Readings in Mathematics (TRIM) |pages=18–37 |publisher=Hindustan Book Agency |doi=10.1007/978-93-86279-56-9_2 |isbn=978-93-80250-51-9 |s2cid=16759683 }} The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.{{cite journal |first=Shun'ichi |last=Amari | title=A foundation of information geometry | journal=Electronics and Communications in Japan |year=1983 |volume=66 |issue=6 |pages=1–10 |doi=10.1002/ecja.4400660602 | url=https://onlinelibrary.wiley.com/doi/abs/10.1002/ecja.4400660602}}

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.{{Cite journal |last=Bauer |first=Martin |last2=Le Brigant |first2=Alice |last3=Lu |first3=Yuxiu |last4=Maor |first4=Cy |date=2024-02-10 |title=The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections |url=https://link.springer.com/article/10.1007/s00526-024-02660-5 |journal=Calculus of Variations and Partial Differential Equations |language=en |volume=63 |issue=2 |pages=56 |doi=10.1007/s00526-024-02660-5 |issn=1432-0835|arxiv=2306.14533 }} 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 descent{{Cite journal |last1=Raskutti |first1=Garvesh |last2=Mukherjee |first2=Sayan |date=March 2015 |title=The Information Geometry of Mirror Descent |url=https://ieeexplore.ieee.org/document/7004065 |journal=IEEE Transactions on Information Theory |volume=61 |issue=3 |pages=1451–1457 |doi=10.1109/TIT.2015.2388583 |arxiv=1310.7780 |issn=0018-9448}} and natural gradient descent{{Cite journal |last1=Abdulkadirov |first1=Ruslan |last2=Lyakhov |first2=Pavel |last3=Nagornov |first3=Nikolay |date=January 2022 |title=Accelerating Extreme Search of Multidimensional Functions Based on Natural Gradient Descent with Dirichlet Distributions |journal=Mathematics |language=en |volume=10 |issue=19 |pages=3556 |doi=10.3390/math10193556 |doi-access=free |issn=2227-7390}}).

The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book, Methods of Information Geometry,{{cite book |first1=Shun'ichi |last1=Amari |first2=Hiroshi |last2=Nagaoka |title=Methods of Information Geometry |series=Translations of Mathematical Monographs |volume=191 |publisher=American Mathematical Society |year=2000 |isbn=0-8218-0531-2 }} and the more recent book by Nihat Ay and others.{{cite book |first1=Nihat |last1=Ay |first2=Jürgen |last2=Jost |author-link2=Jürgen Jost |first3=Hông Vân |last3=Lê |first4=Lorenz |last4=Schwachhöfer |title=Information Geometry |volume=64 |series=Ergebnisse der Mathematik und ihrer Grenzgebiete |publisher=Springer |year=2017 |isbn=978-3-319-56477-7 }} A gentle introduction is given in the survey by Frank Nielsen.{{cite journal|first=Frank |last=Nielsen |journal = Entropy | title=An Elementary Introduction to Information Geometry |date=2018 |url =https://www.mdpi.com/1099-4300/22/10/1100 | volume =22 | number =10 }} In 2018, the journal Information Geometry was released, which is devoted to the field.

{{Cleanup rewrite|Does this list make sense in a general article?|section|date=May 2013}}

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

{{Div col|colwidth=20em}}

• 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

{{Div col end}}

{{Cleanup rewrite|References should be provided for these.|section|date=April 2019}}

As an interdisciplinary field, information geometry has been used in various applications.

Here an incomplete list:

• Statistical inference {{cite book |first1=R. E. |last1=Kass |first2=P. W. |last2=Vos |year=1997 |title=Geometrical Foundations of Asymptotic Inference |series=Series in Probability and Statistics |publisher=Wiley |isbn=0-471-82668-5 }}
• Time series and linear systems
• Filtering problem{{cite journal |last1=Brigo |first1=Damiano |last2=Hanzon |first2=Bernard | last3= LeGland | first3 = Francois | year=1998 |title=A differential geometric approach to nonlinear filtering: the projection filter |journal= IEEE Transactions on Automatic Control |volume= 43 |issue=2 |pages= 247–252 |doi=10.1109/9.661075 |url=https://hal.inria.fr/hal-02101519/file/rr-2598.pdf |author1-link=Damiano Brigo }}
• Quantum systems{{cite journal |last1=van Handel |first1=Ramon |last2=Mabuchi |first2=Hideo |year=2005 |title=Quantum projection filter for a highly nonlinear model in cavity QED | journal= Journal of Optics B: Quantum and Semiclassical Optics| volume= 7 |issue=10 |pages=S226–S236 |doi=10.1088/1464-4266/7/10/005 |arxiv=quant-ph/0503222 |bibcode=2005JOptB...7S.226V |s2cid=15292186 }}
• Neural networks{{cite journal |last1=Zlochin |first1=Mark |last2=Baram |first2=Yoram |year=2001 |title=Manifold Stochastic Dynamics for Bayesian Learning | journal= Neural Computation| volume= 13 |issue=11 |pages=2549–2572 |doi=10.1162/089976601753196021 |pmid=11674851 | url=https://direct.mit.edu/neco/article-abstract/13/11/2549/6467/Manifold-Stochastic-Dynamics-for-Bayesian-Learning?redirectedFrom=fulltext}}
• Machine learning
• Statistical mechanics
• Biology
• Statistics {{cite book |first=Shun'ichi |last=Amari |year=1985 |title=Differential-Geometrical Methods in Statistics |series=Lecture Notes in Statistics |publisher=Springer-Verlag |location=Berlin |isbn=0-387-96056-2 }}{{cite book |first1=M. |last1=Murray |first2=J. |last2=Rice |year=1993 |title=Differential Geometry and Statistics |series=Monographs on Statistics and Applied Probability |volume=48 |publisher=Chapman and Hall |isbn=0-412-39860-5 }}
• Mathematical finance {{cite book |editor-first=Paul |editor-last=Marriott |editor2-first=Mark |editor2-last=Salmon |title=Applications of Differential Geometry to Econometrics |publisher=Cambridge University Press |year=2000 |isbn=0-521-65116-6 }}

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

{{Reflist}}

• [https://www.springer.com/mathematics/geometry/journal/41884] Information Geometry journal by Springer
• [http://bactra.org/notebooks/info-geo.html Information Geometry] overview by Cosma Rohilla Shalizi, July 2010
• [http://math.ucr.edu/home/baez/information/ Information Geometry] notes by John Baez, November 2012
• [http://www.its.caltech.edu/~daw/papers/98-Wage2.pdf Information geometry for neural networks(pdf )], by Daniel Wagenaar

{{Differentiable computing}}