Jacobi fields on statistical manifolds of negative curvature

Physica D: Nonlinear Phenomena

Volumn 234, Issue 1, 2007, Pages 70-80

  Cafaro, C ^a   Ali, S A ^a  

^a STATE UNIVERSITY OF NEW YORK   (United States)

Author keywords

Entropy; Inductive inference; Information geometry; Nonlinear dynamics and chaos; Statistical manifolds

Indexed keywords

CHAOTIC SYSTEMS; COMPUTATIONAL GEOMETRY; DYNAMIC MODELS; DYNAMICS; JACOBIAN MATRICES; STATISTICAL METHODS;

INDUCTIVE INFERENCE; INFORMATION GEOMETRY; JACOBI EQUATION; STATISTICAL MANIFOLDS;

GEODESY;

EID: 34548240536     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2007.07.001     Document Type: Article

References (26)

1. Caticha A. Entropic dynamics. In: Fry R.L. (Ed). Bayesian Inference and Maximum Entropy Methods in Science and Engineering. AIP Conf. Proc. vol. 617 (2002) p. 302
2. Jaynes E.T. Information theory and statistical mechanics I. Phys. Rev. 106 (1957) 620 Information theory and statistical mechanics II, Phys. Rev. 108 (1957) 171
3. Caticha A. Consistency and linearity in quantum theory. Phys. Lett. A244 (1998) 13 Consistency, amplitudes and probabilities in quantum theory, Phys. Rev. A57 (1998) 1572; W. Von der Linden et al. (Eds.), Probability and entropy in quantum theory, in: Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 1999; Insufficient reason and entropy in quantum theory, Found. Phys. 30 (2000) 227
4. Caticha A. Towards a statistical geometrodynamics. In: Elze H.-T. (Ed). Decoherence and Entropy in Complex Systems (2004), Springer Verlag. The information geometry of space and time, in: Presented at the 25th International Workshop on Maximum Entropy and Bayesian Methods in Science and Engineering, San Jose, California. arXiv.org/abs/gr-qc/0508108 August 2005
5. Caticha A. Maximum entropy and Bayesian data analysis: Entropic prior distributions. Phys. Rev. E 70 (2004) 046127
6. Caticha A. Relative entropy and inductive inference. In: Erickson G., and Zhai Y. (Eds). Bayesian Inference and Maximum Entropy Methods in Science and Engineering. AIP Conf. Proc. vol. 707 (2004). p. 75. arXiv.org/abs/physics/0311093
7. A. Caticha, A. Giffin, Updating probabilities, in: Presented at MaxEnt2006, The 26th International Workshop on Bayesian Inference and Maximum Entropy Methods, Paris, France. arXiv:physics/0608185
8. Amari S., and Nagaoka H. Methods of Information Geometry (2000), American Mathematical Society, Oxford University Press
9. Amari S. Differential-geometric methods in statistics. Lecture Notes in Statistics vol. 28 (1985), Springer-Verlag
10. Fisher R.A. Theory of statistical estimation. Proc. Cambridge Philos. Soc. 122 (1925) 700-725
11. Rao C.R. Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 (1945) 81-91
12. Chentsov (Cencov) N.N. Statistical decision rules and optimal inference. Transl. Math. Monographs vol. 53 (1981), Amer. Math. Soc., Providence, RI Originally published in Russian, Nauka, Moscow, 1972
13. Campbell L.L. An extended Cencov characterization of the information metric. Proc. Amer. Math. Soc. 98 (1986) 135
14. Arnold V.I. Mathematical Methods of Classical Physics (1989), Springer-Verlag New York Inc.
15. Zurek W.H., and Paz J.P. Decoherence, chaos, and the second law. Phys. Rev. Lett. 72 (1994) 2508 Quantum Chaos: A decoherent definition, Physica D 83 (1995), 300
16. Eckmann J.P., and Ruelle D. Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57 (1985) 617-656
17. De Felice F., and Clarke C.J.S. Relativity on curved manifolds (1990), Cambridge University Press
18. Misner C.W., Thorne K.S., and Wheeler J.A. Gravitation (1973), Freeman & Co., San Francisco
19. Chicone C., and Mashhoon B. The generalized Jacobi equation. Classical Quantum Gravity 19 (2002) 4231-4248
20. Arwini K., and Dodson C.T.J. Information geometric neighbourhoods of randomness and geometry of the Mckay bivariate gamma 3-manifold. Indian J. Statist. 66 (2004) 211-231
21. Casetti L., Livi R., and Pettini M. Gaussian model for chaotic instability of hamiltonian flows. Phys. Rev. Lett. 74 (1995) 375-378 Riemannian theory of Hamiltonian chaos and Lyapunov exponents, Phys. Rev. E 54 (1996), 5969-5984
22. Kawabe T. Indicator of chaos based on the Riemannian geometric approach. Phys. Rev. E 71 (2005) 017201
23. Caves C.M., and Schack R. Unpredictability, information, and chaos. Complexity 3 1 (1997) 46
24. A.J. Scott, T.A. Brun, C.M. Caves, R. Schack, Hypersensitivity and chaos signatures in the quantum baker's map. arXiv:quant-ph/0606102, 2006
25. Efron B. Defining the curvature of a statistical problem. Ann. Statist. 3 (1975) 1189
26. Rodriguez C.C. Are we cruising a hypothesis space?. In: Von der Linden W., Dose V., Fischer R., and Press R. (Eds). Maximum Entropy and Bayesian Methods (1999), Kluwer, Dordrecht