# nLab Fisher metric

Fisher metric is a metric appearing in information geometry, see there for more information and references. There are also several quantum versions. One is the Bures metric and another is quantum Fisher information matrix.

Most references about Fisher metric are at information geometry, here we add few on quantum and $C^*$-algebraic analogues.

• P. Facchi, R. Kulkarni, V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Ventriglia, Classical and quantum Fisher information in the geometrical formulation of quantum mechanics, Physics Letters A 374 pp. 4801 (2010)doi

There is an analogue in free probability

• D.-V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V: Noncommutative Hilbert transforms, Invent. Math. 132:1 (1998) 189–227.

For singular statistical models (including those arising in machine learning) one needs more version of Fisher metric beyond manifolds; one possibility is in the framework of diffeologies,

• Hông Vân Lê, Diffeological statistical models and diffeological Hausdorff measures, video yt, slides pdf
• Hông Vân Lê, Alexey A. Tuzhilin, Nonparametric estimations and the diffeological Fisher metric, arXiv:2011.13418
• Hông Vân Lê, Diffeological statistical models,the Fisher metric and probabilistic mappings, Mathematics 2020, 8(2),167, arXiv:1912.02090

Last revised on August 16, 2021 at 06:56:04. See the history of this page for a list of all contributions to it.