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Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces. (English) Zbl 1420.62017

Summary: We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher-Rao information metric and the \(\alpha\)-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.

MSC:

62B10 Statistical aspects of information-theoretic topics
62F12 Asymptotic properties of parametric estimators
53A15 Affine differential geometry
53B05 Linear and affine connections
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
94A15 Information theory (general)
94A17 Measures of information, entropy
54E35 Metric spaces, metrizability
54E70 Probabilistic metric spaces
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References:

[1] Amari, S. and Nagaoka, H., Methods of Information Geometry (Oxford University Press, 2000). · Zbl 0960.62005
[2] Amari, S., Differential-Geometric Methods in Statistics, Vol. 28 of (Springer-Verlag, 1985). · Zbl 0559.62001
[3] Amari, S., Information Geometry and Its Applications (Springer- Japan, 2016). · Zbl 1350.94001
[4] Caticha, A., Entropic Inference and the Foundations of Physics; USP Press: São Paulo, Brazil, 2012; Available online: http://www.albany.edu/physics/ACaticha-EIFP-book.pdf.
[5] C. Cafaro, The Information Geometry of Chaos, Ph.D. Thesis in Physics, State University of New York, Albany, NY, USA (2008). · Zbl 1160.81385
[6] Caticha, A., Entropic dynamics, AIP Conf. Proc.617 (2002) 302. · Zbl 1220.37077
[7] Caticha, A. and Giffin, A., Updating probabilities, AIP Conf. Proc.872 (2006) 31.
[8] Giffin, A. and Caticha, A., Updating probabilities with data and moments, AIP Conf. Proc.954 (2007) 74.
[9] A. Giffin, Maximum Entropy: The Universal Method for Inference, Ph.D. Thesis in Physics, State University of New York, Albany, NY, USA (2008).
[10] Caticha, A. and Cafaro, C., From information geometry to Newtonian dynamics, in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conf. Proc.954 (2007) 165.
[11] D. Felice and N. Ay, Dynamical Systems Induced by Canonical Divergence in Dually flat Manifolds, preprint (2018), arXiv:math-ph/1812.04461.
[12] Jaynes, E. T., Macroscopic prediction, in Complex Systems-Operational Approaches in Neurobiology, Physics, and Computers, ed. Haken, H. (Springer-Verlag, Berlin, 1985), pp. 254-269.
[13] Dewar, R. C., Maximum entropy production as an inference algorithm that translates physical assumptions into macroscopic predictions: Don’t shoot the messenger, Entropy11 (2009) 931.
[14] Giffin, A., Cafaro, C. and Ali, S. A., Application of the maximum relative entropy method to the physics of ferromagnetic materials, Physica A455 (2016) 11. · Zbl 1400.82305
[15] Cafaro, C. and Ali, S. A., Maximum caliber inference and the stochastic Ising model, Phys. Rev. E94 (2016) 052145.
[16] Ali, S. A. and Cafaro, C., Theoretical investigations of an information geometric approach to complexity, Rev. Math. Phys.29 (2017) 1730002. · Zbl 1431.94038
[17] Felice, D., Cafaro, C. and Mancini, S., Information geometric methods for complexity, Chaos28 (2018) 032101. · Zbl 1407.81041
[18] Ali, S. A., Cafaro, C., Gassner, S. and Giffin, A., An information geometric perspective on the complexity of macroscopic predictions arising from incomplete information, Adv. Math. Phys.2018 (2018), Article ID: 2048521. · Zbl 1410.82012
[19] Misner, C. W., Thorne, K. S. and Wheeler, J. A., Gravitation, W. H. Freeman and Company (1973).
[20] Cencov, N. N., Statistical Decision Rules and Optimal Inference, , Vol. 53 (Amer. Math. Soc., Providence-RI, 1981).
[21] Campbell, L. L., An extended Cencov characterization of the information metric, Proc. Am. Math. Soc.98 (1986) 135. · Zbl 0608.62013
[22] Peter, A. and Rangarajan, A., A New Closed-Form Information Metric for Shape Analysis, First MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical, Statistical and Registration Methods for Modeling Biological Shape Variability, October 2006, Copenhagen, (Denmark, 2006), pp. 100-101.
[23] Havrda, J. and Charvat, F., Quantification method of classification processes. Concept of structural \(\alpha \)-entropy, Kybernetika3 (1967) 30. · Zbl 0178.22401
[24] Burbea, J. and Rao, C. R., Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, J. Multivariate Anal.12 (1982) 575. · Zbl 0526.60015
[25] Cafaro, C. and Ali, S. A., Jacobi fields on statistical manifolds of negative curvature, Physica D234 (2007) 70. · Zbl 1129.53019
[26] Cafaro, C. and Mancini, S., Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields, Physica D240 (2011) 607. · Zbl 1217.37079
[27] Casetti, L., Pettini, M. and Cohen, E. G. D., Geometric approach to Hamiltonian dynamics and statistical mechanics, Phys. Rep.337 (2000) 237.
[28] De Felice, F. and Clarke, C. J. S., Relativity on Curved Manifolds (Cambridge University Press, 1990). · Zbl 0705.53001
[29] Weinberg, S., Gravitation and Cosmology (John Wiley & Sons Inc., 1972).
[30] Nagle, R. K., Saff, E. B. and Snider, A. D., Fundamentals of differential equations (Pearson Education Inc., 2012). · Zbl 0949.34001
[31] Peres, A. and Terno, D. R., Quantum information and relativity theory, Rev. Mod. Phys.76 (2004) 93. · Zbl 1205.81050
[32] Fidkowski, L., Hubeny, V., Kleban, M. and Shenker, S., The black hole singularity in AdS/CFT, J. High Energy Phys.02 (2004) 014.
[33] V. Hubeny, Black Hole Singularity in AdS/CFT, preprint (2004), arXiv:hep-th/0401138 (2004).
[34] Hildebrand, F. B., Introduction to Numerical Analysis (Dover Publications Inc., 1987). · Zbl 0641.65001
[35] Cafaro, C., Geometric algebra and information geometry for quantum computational software, Physica A470 (2017) 154. · Zbl 1400.81044
[36] Cafaro, C. and Alsing, P. M., Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes, Phys. Rev. E97 (2018) 042110.
[37] Kahle, T., Olbrich, E., Jost, J. and Ay, N., Complexity measures from interaction structures, Phys. Rev. E79 (2009) 026201.
[38] Capozziello, S., Lambiase, G. and Stornaiolo, C., Geometric classification of the torsion tensor in space-time, Annalen der Physik10 (2001) 713. · Zbl 0974.83035
[39] Capozziello, S., De Falco, V. and Pincak, R., Torsion in Bianchi IX cosmology, Int. J. Geom. Meth. Mod. Phys.14 (2017) 1750186. · Zbl 1386.83035
[40] Capozziello, S. and Luongo, O., Dark energy from entanglement entropy, Int. J. Theor. Phys.52 (2013) 2698. · Zbl 1274.83144
[41] Capozziello, S. and Luongo, O., Entangled states in quantum cosmology and the interpretation of Lambda, Entropy13 (2011) 528. · Zbl 1229.83075
[42] Capozziello, S., Luongo, O. and Mancini, S., Cosmological dark energy effects from entanglement, Phys. Lett. A377 (2013) 1061. · Zbl 1274.83144
[43] Kim, D.-H., Ali, S. A., Cafaro, C. and Mancini, S., Information geometric modeling of scattering induced quantum entanglement, Phys. Lett. A375 (2011) 2868.
[44] Kim, D.-H., Ali, S. A., Cafaro, C. and Mancini, S., Information geometry of quantum entangled wave-packets, Physica A391 (2012) 4517.
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