## Publicacions Matemàtiques

### Intrinsic geometry on the class of probability densities and exponential families

#### Abstract

We present a way of thinking of exponential families as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group $G^+$ of the group $G$ of all invertible elements in the algebra $\mathcal{A}$ of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class $\mathcal{D}$ of densities with respect to a given measure will happen to be representatives of equivalence classes defining a projective space in $\mathcal{A}$. The natural geometry is defined by an intrinsic group action which allows us to think of the class of positive, invertible functions $G^+$ as a homogeneous space. Also, the parallel transport in $G^+$ and $\mathcal{D}$ will be given by the original group action. Besides studying some relationships among these constructions, we examine some Riemannian geometries and provide a geometric interpretation of Pinsker's and other classical inequalities. Also we provide a geometric reinterpretation of some relationships between polynomial sequences of convolution type, probability distributions on $\mathbb{N}$ in terms of geodesics in the Banach space $\ell_1(\alpha)$.

#### Article information

Source
Publ. Mat., Volume 51, Number 2 (2007), 309-332.

Dates
First available in Project Euclid: 31 July 2007

https://projecteuclid.org/euclid.pm/1185912165

Mathematical Reviews number (MathSciNet)
MR2334793

Zentralblatt MATH identifier
1141.46025

#### Citation

Gzyl, Henryk; Recht, Lázaro. Intrinsic geometry on the class of probability densities and exponential families. Publ. Mat. 51 (2007), no. 2, 309--332. https://projecteuclid.org/euclid.pm/1185912165

#### References

• S.-I. Amari, “Differential-geometrical methods in statistics”, Lecture Notes in Statistics 28, Springer-Verlag, New York, 1985.
• S.-I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen and C. R. Rao, “Differential geometry in statistical inference”, Institute of Mathematical Statistics Lecture Notes-Monograph Series 10, Institute of Mathematical Statistics, Hayward, CA, 1987.
• O. E. Barndorff-Nielsen, “Information and exponential families in statistical theory”, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1978.
• C. Berg, J. P. R. Christensen and P. Ressel, “Harmonic analysis on semigroups”, Theory of positive definite and related functions, Graduate Texts in Mathematics 100, Springer-Verlag, New York, 1984.
• L. Brown, Sufficient statistics in the case of independent random variables, Ann. Math. Statist 35 (1964), 1456\Ndash1474.
• A. Di Bucchianico, “Probabilistic and analytical aspects of the umbral calculus”, CWI Tract 119, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1997.
• G. Corach, H. Porta and L. Recht, The geometry of the space of selfadjoint invertible elements in a $C\sp *$-algebra, Integral Equations Operator Theory 16(3) (1993), 333\Ndash359.
• B. Efron, The geometry of exponential families, Ann. Statist. 6(2) (1978), 362\Ndash376.
• F. Esscher, On the probability function in the collective theory of risk, Scand. Actuar. J. 15 (1932), 175\Ndash195.
• I. Gelfand, D. Raikov and G. Shilov, “Commutative normed rings”, Translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., New York, 1964.
• P. Gibilisco and G. Pistone, Connections on non\guioparametric statistical manifolds by Orlicz space geometry, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(2) (1998), 325\Ndash347.
• H. Gzyl and L. Recht, A geometry in the space of probabilities II: Projective spaces and exponential families, Rev. Mat. Iberoamericana (to appear).
• E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. (2) 106 (1957), 620\Ndash630.
• S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, Vol. II, Interscience Tracts in Pure and Applied Mathematics 15, Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.
• S. Kullback, “Information theory and statistics”, 2nd. revised edition, Dover Publications, Inc., Mineola, NY, 1968.
• R. J. Larsen and M. L. Marx, “An Introduction to Mathematical Statistics and its Applications”, Prentice-Hall, Englewood Cliffs, N.J., 2006.
• G. Pistone and M. P. Rogantin, The exponential statistical manifold: mean parameters, orthogonality and space transformations, Bernoulli 5(4) (1999), 721\Ndash760.
• G. Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Statist. 23(5) (1995), 1543\Ndash1561.
• H. Porta and L. Recht, Conditional expectations and operator decompositions, Ann. Global Anal. Geom. 12(4) (1994), 335\Ndash339.
• H. Porta and L. Recht, Exponential sets and their geometric motions, J. Geom. Anal. 6(2) (1996), 277\Ndash285.
• R. F. Streater, Classical and quantum info-manifolds, Analytical study of quantum information and related fields (Japanese) (Kyoto, 2000), Sūrikaisekikenkyūsho Kōkyūroku 1196 (2001), 32\Ndash51.
• I. Vajda, “Theory of statistical inference and information”, Theory and Decision Library B 11, Kluwer Academic Publishers, Dordrecht, 1989.
• D. Williams, “Weighing the odds”, A course in probability and statistics, Cambridge University Press, Cambridge, 2001.