July  2022, 42(7): 3593-3627. doi: 10.3934/dcds.2022026

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

1. 

Instituto de Matemática e Estatística - UFRGS, 91509-900, Porto Alegre, Brazil

2. 

Departamento Interdisciplinar - UFRGS, 95590-000, Tramandaí, Brazil

* Corresponding author: Artur O. Lopes

Received  June 2021 Revised  November 2021 Published  July 2022 Early access  March 2022

Fund Project: Artur O. Lopes would like to acknowledge financial support by CNPq

It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an a priori probability kernel $ \hat{\nu} $ and a probability $ \pi $ on the measurable space $ X\times Y $ we consider an appropriate definition of entropy of $ \pi $ relative to $ \hat{\nu} $, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case $ \hat{\nu} $ is identified with a probability $ \nu $ on $ X $. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). Via the concepts of involution kernel and dual potential, one can ask if a given potential is symmetric - the relevant information is available in the potential. In the affirmative case, its corresponding equilibrium state has zero entropy production.

Citation: Artur O. Lopes, Jairo K. Mengue. On information gain, Kullback-Leibler divergence, entropy production and the involution kernel. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3593-3627. doi: 10.3934/dcds.2022026
References:
[1]

D. AguiarL. Cioletti and R. Ruviaro, A variational principle for the specific entropy for symbolic systems with uncountable alphabets, Math. Nachr., 291 (2018), 2506-2515.  doi: 10.1002/mana.201700229.

[2]

A. Baraviera, R. Leplaideur and A. Lopes, Ergodic Optimization, Zero Temperature and the Max-Plus Algebra, 29$^{\text{o}}$ Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, 2013.

[3]

A. BaravieraA. O. Lopes and P. Thieullen, A large deviation principle for equilibrium states of Hölder potencials: The zero temperature case, Stochastics and Dynamics, 6 (2006), 77-96.  doi: 10.1142/S0219493706001657.

[4]

T. BenoistV. JakšićY. Pautrat and C-A. Pillet, On entropy production of repeated quantum measurements I. General theory, Comm. Math. Phys., 357 (2018), 77-123.  doi: 10.1007/s00220-017-2947-1.

[5]

M. Capinski and E. Kopp, Measure Integral and Probability, Springer-Verlag, 2004. doi: 10.1007/978-1-4471-0645-6.

[6]

J-R. Chazottes, E. Floriani and R. Lima, Relative entropy and identification of Gibbs measures in dynamical systems, J. Statist. Phys., 90 (1998), 679–725. doi: 10.1023/A:1023220802597.

[7]

L. Cioletti, L. Melo, R. Ruviaro and E. A. Silva, On the dimension of the space of harmonic functions on transitive shift spaces, Advances in Math, 385 (2021), Article 1077585. doi: 10.1016/j.aim.2021.107758.

[8]

L. CiolettiM. DenkerA. O. Lopes and M. Stadlbauer, Spectral properties of the Ruelle operator for product-type potentials on shift spaces, J. Lond. Math. Soc. (2), 95 (2017), 684-704.  doi: 10.1112/jlms.12031.

[9]

L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455.  doi: 10.1007/s10955-015-1202-4.

[10]

L. Cioletti and A. O. Lopes, Correlation inequalities and monotonicity properties of the Ruelle operator, Stochastics and Dynamics, 19 (2019), 1950048, 31 pp. doi: 10.1142/S0219493719500485.

[11]

G. Contreras, A. O. Lopes and E. R. Oliveira, Ergodic transport theory, periodic maximizing probabilities and the twist condition, "Modeling, Optimization, Dynamics and Bioeconomy I", Springer Proceedings in Mathematics and Statistics, Volume 73, Edit. David Zilberman and Alberto Pinto, (2014), 183–219. doi: 10.1007/978-3-319-04849-9_12.

[12]

G. ContrerasA. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Erg. Theo. and Dyn. Syst., 21 (2001), 1379-1409.  doi: 10.1017/S0143385701001663.

[13]

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2 ed. Wiley-Interscience, 2006.

[14]

G. Crooks, Entropy production fluctuation Theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E, 60 (1999), 2721.  doi: 10.1103/PhysRevE.60.2721.

[15]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North-Holland, 1976.

[16]

A. M. Fisher and A. Lopes, Exact bounds for the polynomial decay of correlation, 1/f noise and the central limit Theorem for a non-Holder Potential, Nonlinearity, 14 (2001), 1071-1104.  doi: 10.1088/0951-7715/14/5/310.

[17]

G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., 74 (1995), 2694.  doi: 10.1103/PhysRevLett.74.2694.

[18]

H.-O. Georgii, Gibbs Measures and Phase Transitions, De Gruyter Studies in Mathematics, 9, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329.

[19]

R. M. Gray, Entropy and Information Theory, New York, Springer-Verlag, 1990. doi: 10.1007/978-1-4757-3982-4.

[20]

L. Y. Hataishi, Spectral Triples em Formalismo Termodinâmicoe Kernel de Involução Para Potenciais Walters, Master Dissertation, Pos. Grad. Mat - UFRGS, 2020.

[21]

L. Y. Hataishi and A. O. Lopes, The dual potential for functions on the Walters' family, to appear.

[22]

F. Hofbauer, Examples for the nonuniquenes of the equilibrium state, Transactions AMS, 228 (1977), 223–241. doi: 10.1090/S0002-9947-1977-0435352-1.

[23]

D.-Q. JiangM. Qian and M.-P. Qian, Entropy production and information gain in axiom-a systems, Commun. Math. Phys., 214 (2000), 389-409.  doi: 10.1007/s002200000277.

[24]

S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statist., 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.

[25]

A. O. Lopes, The Zeta function, non-differentiability of pressure and the critical exponent of transition, Advances in Math., 101 (1993), 133-167.  doi: 10.1006/aima.1993.1045.

[26]

A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Erg. Theo. Dyn. Sys., 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15.

[27]

A. O. Lopes and J. K. Mengue, Thermodynamic formalism for Haar systems in noncommutative integration: Probability kernels and entropy of transverse measures, Erg. Theo. Dyn. Sys., 41 (2021), 1835-1863.  doi: 10.1017/etds.2020.24.

[28]

A. O. Lopes, J. K. Mengue, J. Mohr and C. G. Moreira, Large deviations for quantum spin probabilities at temperature zero, Stochastics and Dynamics, 18 (2018), 1850044, 26 pp. doi: 10.1142/S0219493718500442.

[29]

A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy, pressure and duality for Gibbs plans in ergodic transport, Bull. Braz. Math. Soc., 46 (2015), 353-389.  doi: 10.1007/s00574-015-0095-9.

[30]

A. O. Lopes, E. R. Oliveira and Ph. Thieullen, The dual potential, the involution kernel and transport in ergodic optimization, Dynamics, Games and Science -International Conference and Advanced School Planet Earth DGS II, Portugal (2013), Edit. J-P Bourguignon, R. Jelstch, A. Pinto and M. Viana, Springer Verlag, (2015), 357–398. doi: 10.1007/978-3-319-16118-1_20.

[31]

A. O. Lopes and R. Ruggiero, Nonequilibrium in thermodynamic formalism: The second law, gases and information geometry, Qual. Theo. of Dyn. Syst., 21 (2022), 1-44.  doi: 10.1007/s12346-021-00551-0.

[32]

C. Maes, The fluctuation theorem as a Gibbs property, J. Statist. Phys., 95 (1999), 367-392.  doi: 10.1023/A:1004541830999.

[33]

R. J. McEliece, The Theory of Information and Coding, Addison-Wesley, 1977.

[34]

L. C. Melo, On the Maximal Eigenspace of the Ruelle Operator, PhD Thesis. UNB (2020), (available online from: https://repositorio.unb.br/handle/10482/39599).

[35]

J. Mengue, Tópicos de álgebra linear e probabilidade, SBM, (2016).

[36]

J. K. Mengue and E. R. Oliveira, Duality results for iterated function systems with a general family of branches, Stochastics and Dynamics, 17 (2017), 1750021, 23 pp. doi: 10.1142/S0219493717500216.

[37]

J. Mohr, Product type potential on the XY model: Selection of maximizing probability and a large deviation principle, to appear in Qual. Theo. of Dyn. Syst.

[38]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.

[39]

J. R. Quinlan, Induction of decision trees, Machine Learning, 1 (1986), 81-106.  doi: 10.1007/BF00116251.

[40]

D. Ruelle, A generalized detailed balance relation, J. Stat. Phys., 164 (2016), 463-471.  doi: 10.1007/s10955-016-1564-2.

[41]

C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 27 (1948), 379-423.  doi: 10.1002/j.1538-7305.1948.tb01338.x.

[42]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Press, 2016. doi: 10.1017/CBO9781316422601.

[43]

P. Walters, A natural space of functions for the Ruelle operator theorem, Erg. Theo. Dyn. Syst., 27 (2007), 1323-1348.  doi: 10.1017/S0143385707000028.

[44]

P. Walters, An introduction to Ergodic Theory, Springer Verlag, 1982.

show all references

References:
[1]

D. AguiarL. Cioletti and R. Ruviaro, A variational principle for the specific entropy for symbolic systems with uncountable alphabets, Math. Nachr., 291 (2018), 2506-2515.  doi: 10.1002/mana.201700229.

[2]

A. Baraviera, R. Leplaideur and A. Lopes, Ergodic Optimization, Zero Temperature and the Max-Plus Algebra, 29$^{\text{o}}$ Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, 2013.

[3]

A. BaravieraA. O. Lopes and P. Thieullen, A large deviation principle for equilibrium states of Hölder potencials: The zero temperature case, Stochastics and Dynamics, 6 (2006), 77-96.  doi: 10.1142/S0219493706001657.

[4]

T. BenoistV. JakšićY. Pautrat and C-A. Pillet, On entropy production of repeated quantum measurements I. General theory, Comm. Math. Phys., 357 (2018), 77-123.  doi: 10.1007/s00220-017-2947-1.

[5]

M. Capinski and E. Kopp, Measure Integral and Probability, Springer-Verlag, 2004. doi: 10.1007/978-1-4471-0645-6.

[6]

J-R. Chazottes, E. Floriani and R. Lima, Relative entropy and identification of Gibbs measures in dynamical systems, J. Statist. Phys., 90 (1998), 679–725. doi: 10.1023/A:1023220802597.

[7]

L. Cioletti, L. Melo, R. Ruviaro and E. A. Silva, On the dimension of the space of harmonic functions on transitive shift spaces, Advances in Math, 385 (2021), Article 1077585. doi: 10.1016/j.aim.2021.107758.

[8]

L. CiolettiM. DenkerA. O. Lopes and M. Stadlbauer, Spectral properties of the Ruelle operator for product-type potentials on shift spaces, J. Lond. Math. Soc. (2), 95 (2017), 684-704.  doi: 10.1112/jlms.12031.

[9]

L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455.  doi: 10.1007/s10955-015-1202-4.

[10]

L. Cioletti and A. O. Lopes, Correlation inequalities and monotonicity properties of the Ruelle operator, Stochastics and Dynamics, 19 (2019), 1950048, 31 pp. doi: 10.1142/S0219493719500485.

[11]

G. Contreras, A. O. Lopes and E. R. Oliveira, Ergodic transport theory, periodic maximizing probabilities and the twist condition, "Modeling, Optimization, Dynamics and Bioeconomy I", Springer Proceedings in Mathematics and Statistics, Volume 73, Edit. David Zilberman and Alberto Pinto, (2014), 183–219. doi: 10.1007/978-3-319-04849-9_12.

[12]

G. ContrerasA. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Erg. Theo. and Dyn. Syst., 21 (2001), 1379-1409.  doi: 10.1017/S0143385701001663.

[13]

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2 ed. Wiley-Interscience, 2006.

[14]

G. Crooks, Entropy production fluctuation Theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E, 60 (1999), 2721.  doi: 10.1103/PhysRevE.60.2721.

[15]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North-Holland, 1976.

[16]

A. M. Fisher and A. Lopes, Exact bounds for the polynomial decay of correlation, 1/f noise and the central limit Theorem for a non-Holder Potential, Nonlinearity, 14 (2001), 1071-1104.  doi: 10.1088/0951-7715/14/5/310.

[17]

G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., 74 (1995), 2694.  doi: 10.1103/PhysRevLett.74.2694.

[18]

H.-O. Georgii, Gibbs Measures and Phase Transitions, De Gruyter Studies in Mathematics, 9, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329.

[19]

R. M. Gray, Entropy and Information Theory, New York, Springer-Verlag, 1990. doi: 10.1007/978-1-4757-3982-4.

[20]

L. Y. Hataishi, Spectral Triples em Formalismo Termodinâmicoe Kernel de Involução Para Potenciais Walters, Master Dissertation, Pos. Grad. Mat - UFRGS, 2020.

[21]

L. Y. Hataishi and A. O. Lopes, The dual potential for functions on the Walters' family, to appear.

[22]

F. Hofbauer, Examples for the nonuniquenes of the equilibrium state, Transactions AMS, 228 (1977), 223–241. doi: 10.1090/S0002-9947-1977-0435352-1.

[23]

D.-Q. JiangM. Qian and M.-P. Qian, Entropy production and information gain in axiom-a systems, Commun. Math. Phys., 214 (2000), 389-409.  doi: 10.1007/s002200000277.

[24]

S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statist., 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.

[25]

A. O. Lopes, The Zeta function, non-differentiability of pressure and the critical exponent of transition, Advances in Math., 101 (1993), 133-167.  doi: 10.1006/aima.1993.1045.

[26]

A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Erg. Theo. Dyn. Sys., 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15.

[27]

A. O. Lopes and J. K. Mengue, Thermodynamic formalism for Haar systems in noncommutative integration: Probability kernels and entropy of transverse measures, Erg. Theo. Dyn. Sys., 41 (2021), 1835-1863.  doi: 10.1017/etds.2020.24.

[28]

A. O. Lopes, J. K. Mengue, J. Mohr and C. G. Moreira, Large deviations for quantum spin probabilities at temperature zero, Stochastics and Dynamics, 18 (2018), 1850044, 26 pp. doi: 10.1142/S0219493718500442.

[29]

A. O. LopesJ. K. MengueJ. Mohr and R. R. Souza, Entropy, pressure and duality for Gibbs plans in ergodic transport, Bull. Braz. Math. Soc., 46 (2015), 353-389.  doi: 10.1007/s00574-015-0095-9.

[30]

A. O. Lopes, E. R. Oliveira and Ph. Thieullen, The dual potential, the involution kernel and transport in ergodic optimization, Dynamics, Games and Science -International Conference and Advanced School Planet Earth DGS II, Portugal (2013), Edit. J-P Bourguignon, R. Jelstch, A. Pinto and M. Viana, Springer Verlag, (2015), 357–398. doi: 10.1007/978-3-319-16118-1_20.

[31]

A. O. Lopes and R. Ruggiero, Nonequilibrium in thermodynamic formalism: The second law, gases and information geometry, Qual. Theo. of Dyn. Syst., 21 (2022), 1-44.  doi: 10.1007/s12346-021-00551-0.

[32]

C. Maes, The fluctuation theorem as a Gibbs property, J. Statist. Phys., 95 (1999), 367-392.  doi: 10.1023/A:1004541830999.

[33]

R. J. McEliece, The Theory of Information and Coding, Addison-Wesley, 1977.

[34]

L. C. Melo, On the Maximal Eigenspace of the Ruelle Operator, PhD Thesis. UNB (2020), (available online from: https://repositorio.unb.br/handle/10482/39599).

[35]

J. Mengue, Tópicos de álgebra linear e probabilidade, SBM, (2016).

[36]

J. K. Mengue and E. R. Oliveira, Duality results for iterated function systems with a general family of branches, Stochastics and Dynamics, 17 (2017), 1750021, 23 pp. doi: 10.1142/S0219493717500216.

[37]

J. Mohr, Product type potential on the XY model: Selection of maximizing probability and a large deviation principle, to appear in Qual. Theo. of Dyn. Syst.

[38]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.

[39]

J. R. Quinlan, Induction of decision trees, Machine Learning, 1 (1986), 81-106.  doi: 10.1007/BF00116251.

[40]

D. Ruelle, A generalized detailed balance relation, J. Stat. Phys., 164 (2016), 463-471.  doi: 10.1007/s10955-016-1564-2.

[41]

C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 27 (1948), 379-423.  doi: 10.1002/j.1538-7305.1948.tb01338.x.

[42]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Press, 2016. doi: 10.1017/CBO9781316422601.

[43]

P. Walters, A natural space of functions for the Ruelle operator theorem, Erg. Theo. Dyn. Syst., 27 (2007), 1323-1348.  doi: 10.1017/S0143385707000028.

[44]

P. Walters, An introduction to Ergodic Theory, Springer Verlag, 1982.

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