# References for use of symplectic geometry in statistics?

I have heard that many problems in Mathematical Statistics can be stated and solved in terms of Symplectic Geometry. Unfortunately this was a pretty vague statement and I am interested in something more concrete. Also there seem to be some books written form this perspective, but I couldn't find any.

• Where did you hear this statement? – whuber Dec 12 '10 at 18:20

## 3 Answers

I know nothing whatsoever about symplectic geometry, but a bit of googling brought up a 1997 article in the Journal of Statistical Planning & Inference by Barndorff-Nielsen & Jupp, which contains this quote:

Some other links between statistics and symplectic geometry have been discussed by Friedrich and Nakamura. Friedrich (1991) established some connections between expected (Fisher) information and symplectic structures. However, his approach and results are quite different from those considered here. Nakamura (1993, 1994) has shown that certain parametric statistical models in which the parameter space M is an even-dimensional vector space (and so has the symplectic structure of the cotangent space of a vector space) give rise to completely integrable Hamiltonian systems on M.

The cited refs are:

• Friedrich, T., 1991. Die Fisher-lnformation und symplectische Strukturen. Math. Nachr. 153: 273-296.
• Nakamura, Y., 1993. Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions. Japan. J. Ind. Appl. Math. 10: 179-189.
• Nakamura, Y., 1994. Gradient systems associated with probability distributions. Japan. J. Ind. Appl. Math. 11: 21-30.

The article's Introduction says B-N & others have used differential geometry as an approach to statistical asymptotics. Symplectic geometry is a branch of differential geometry (according to Wikipedia). A Google Books search finds several books about the application of differential geometry to statistics and related fields such as econometrics.

• +1 Good finds. However, most of the differential geometry applications to statistics do not use symplectic geometry. I think our difficulty in finding references is testimony to how obscure and rare are the applications of symplectic geometry to statistics. – whuber Dec 12 '10 at 21:45

A direct connection would be unexpected: the two fields appear to have little in common. For example, a modern introduction to symplectic geometry published by the American Mathematical Society appears to make no mention of mathematical statistics at all.

At best it seems any connection would come through mathematical physics. A symplectic geometry on phase space naturally arises in the Hamiltonian formulation of classical mechanics and that in turn can be used to explore global properties of physical systems. The study of periodic and near-periodic orbits becomes somewhat statistical (e.g., ergodic theorems). When applied to a system with many degrees of freedom it would conceivably relate some aspects of symplectic geometry to thermodynamics, which is inherently a statistical theory

Symplectic model of Statistical Physics and Information Geometry is given by Souriau model of "Lie groups Thermodynamics": Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics,
Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation,
(Souriau-Casimir Lie Groups Thermodynamics & Machine Learning: A slide presentation)(https://franknielsen.github.io/SPIG-LesHouches2020/Barbaresco-SPILG2020.pdf),
the youtube version

• please add references for your links in case they die in the future – Antoine Aug 12 '20 at 6:31