Asymmetric Topologies on Statistical Manifolds

详细信息    查看全文

• 刊名：Lecture Notes in Computer Science
• 出版年：2015
• 出版时间：2015
• 年：2015
• 卷：9389
• 期：1
• 页码：203-210
• 全文大小：174 KB
• 参考文献：1.Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)MathSciNet CrossRef MATH
  2.Chentsov, N.N.: A nonsymmetric distance between probability distributions, entropy and the Pythagorean theorem. Math. notes Acad. Sci. USSR 4(3), 323–332 (1968)
  3.Chentsov, N.N.: Statistical decision rules and optimal inference. Nauka, Moscow, U.S.S.R. In: Russian, English translation: Providence, p. 1982. AMS, RI (1972)
  4.Morozova, E.A., Chentsov, N.N.: Natural geometry of families of probability laws. In: Prokhorov, Y.V. (ed.) Probability Theory 8. Volume 83 of Itogi Nauki i Tekhniki, pp. 133–265. VINITI, Russian (1991)
  5.Fletcher, P., Lindgren, W.F.: Quasi-uniform spaces. Lecture notes in pure and applied mathematics. Vol. 77, Marcel Dekker, New York (1982)
  6.Cobzas, S.: Functional Analysis in Asymmetric Normed Spaces. Birkhäuser, Basel (2013)CrossRef MATH
  7.Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23(5), 1543–1561 (1995)MathSciNet CrossRef MATH
  8.Bernoulli, D.: Commentarii academiae scientarum imperialis petropolitanae. Econometrica 22, 23–36 (1954)MathSciNet CrossRef
  9.Asplund, E., Rockafellar, R.T.: Gradients of convex functions. Trans. Am. Math. Soc. 139, 443–467 (1969)MathSciNet CrossRef MATH
  10.Moreau, J.J.: Functionelles Convexes, Lectrue Notes, Séminaire sur les équations aux derivées partielles. Collége de France, Paris (1967)
  11.Borodin, P.A.: The Banach-Mazur theorem for spaces with asymmetric norm. Math. Notes 69(3–4), 298–305 (2001)MathSciNet CrossRef MATH
  12.Tikhomirov, V.M.: Convex analysis. In: Gamkrelidze, R.V. (ed.) Analysis II. Encyclopedia of Mathematical Sciences. vol. 14, pp. 1–92. Springer-Verlag, Heidelberg (1990)
  13.Rockafellar, R.T.: Conjugate duality and optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 16, Society for Industrial and Applied Mathematics, PA (1974)
  14.García-Raffi, L.M., Romaguera, S., Sánchez-Pérez, E.A.: On Hausdorff asymmetric normed linear spaces. Houston J. Math. 29(3), 717–728 (2003)MathSciNet MATH
  15.Krasnoselskii, M.A., Rutitskii, Y.B.: Convex functions and Orlicz spaces. Fizmatgiz, Moscow (1958). English translation.: P. Noordhoff, Ltd., Groningen (1961)
  16.Reilly, I.L., Subrahmanyam, P.V., Vamanamurthy, M.K.: Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte für Mathematik 93, 127–140 (1982)MathSciNet CrossRef MATH
  
• 作者单位：Roman V. Belavkin (15)
  
  15. School of Science and Technology, Middlesex University, London, NW4 4BT, UK
  
• 丛书名：Geometric Science of Information
• ISBN：978-3-319-25040-3
• 刊物类别：Computer Science
• 刊物主题：Artificial Intelligence and Robotics
  Computer Communication Networks
  Software Engineering
  Data Encryption
  Database Management
  Computation by Abstract Devices
  Algorithm Analysis and Problem Complexity
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1611-3349

文摘

Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.