Skip to main content

Information Theory at the Service of Science

  • Chapter
Entropy, Search, Complexity

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 16))

Abstract

Information theory is becoming more and more important for many fields. This is true for engineering- and technology-based areas but also for more theoretically oriented sciences such as probability and statistics.

Research supported by INTAS, project 00-738, and by the Danish Natural Science Research Council.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. I. Amari, Information geometry on hierarchy of probability distributions, IEEE Trans. Inform. Theory, 47 (2001), 1701–1711.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Applebaum, Probability and Information. An integrated approach, Cambridge Univ. Press (Cambridge, 1996).

    MATH  Google Scholar 

  3. J. P. Aubin, Optima and equilibria. An introduction to nonlinear analysis, Springer (Berlin, 1993).

    MATH  Google Scholar 

  4. A. R. Barron, Entropy and the central limit theorem, Ann. Probab., 14 (1) (1986), 336–342.

    MATH  MathSciNet  Google Scholar 

  5. A. R. Barron and O. Johnson, Fisher information inequalities and the central limit theorem, submitted for publication, Probab. Theory Relat. Fields, 129 (2004), 391–409.

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Bellini and M. Frittelli, On the existence of minimax martingale measures, Mathematical Finance, 12 (2002), 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. and Math. Phys., 7 (1967), 200–217. Translated from Russian.

    Article  Google Scholar 

  8. M. Broom, Using game theory to model the evolution of information: An illustrative game, Entropy, 4 (2002), 35–46. Online at http://www.unibas.ch/mdpi/entropy/.

    MATH  MathSciNet  Google Scholar 

  9. N. N. Čencov, A nonsymmetric distance between probability distributions, entropy and the Pythagorean theorem. Math. Zametki, 4 (1968), 323–332 (in Russian).

    Google Scholar 

  10. N. N. Čencov, Statistical decision rules and optimal inference, Nauka (Moscow, 1972), in Russian, translation in “Translations of Mathematical Monographs”, 53. American Mathematical Society (1982).

    Google Scholar 

  11. T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley (New York, 1991).

    MATH  Google Scholar 

  12. I. Csiszár, Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitverteilung, A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 7 (1962), 137–158.

    MATH  Google Scholar 

  13. I. Csiszár, A class of measures of informativity of observation channels, Period. Math. Hungar., 2 (1972), 191–213.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Csiszár, I-divergence geometry of probability distributions and minimization problems, Ann. Probab., 3 (1975), 146–158.

    MATH  Google Scholar 

  15. I. Csiszár, Sanov property, generalized I-projection and a conditional limit theorem, Ann. Probab., 12 (1984), 768–793.

    MATH  MathSciNet  Google Scholar 

  16. I. Csiszár, Why least squares and maximum entropy? an axiomatic approach to inference for linear inverse problems, Ann. Stat., 19 (1991), 2032–2066.

    MATH  Google Scholar 

  17. I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic (New York, 1981).

    MATH  Google Scholar 

  18. I. Csiszár and F. Matúš, Convex cores of measures on ℝd, Stud. Sci. Math. Hungar., 38 (2001), 177–190.

    MATH  Google Scholar 

  19. I. Csiszár and F. Matúš, Information projections revisited, IEEE Trans. Inform. Theory, 49 (2003), 1474–1490.

    Article  MATH  MathSciNet  Google Scholar 

  20. L. D. Davisson and A. Leon-Garcia, A source matching approach to finding minimax codes, IEEE Trans. Inform. Theory, 26 (1980), 166–174.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Delbaen, P. Grandits, T. Rheinlaender, D. Samperi, M. Schweizer and C. Stricker, Exponential hedging and entropic penalties, Mathematical Finance, 12 (2002), 99–123.

    Article  MATH  MathSciNet  Google Scholar 

  22. B. de Finetti, Theory of Probability, Wiley (London, 1974). Italian original 1970.

    MATH  Google Scholar 

  23. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Bartlett Publishers International, Boston, 1993.

    MATH  Google Scholar 

  24. M. J. Donald, On the relative entropy. Commun. Math. Phys., 105 (1985), 13–34.

    Article  MathSciNet  Google Scholar 

  25. Z. Drezner and H. Hamacher, editors, Facility location. Applications and Theory, Springer (Berlin, 2002).

    MATH  Google Scholar 

  26. P. D. Grünwald and A. P. Dawid, Game theory, maximum entropy, minimum discrepancy, and robust bayesian decision theory, Ann. Stat., 32 (2004), 1367–1433.

    Article  MATH  Google Scholar 

  27. P. Harremoës, Binomial and Poisson distributions as maximum entropy distributions. IEEE Trans. Inform. Theory, 47 (5) (July 2001), 2039–2041.

    Article  MATH  MathSciNet  Google Scholar 

  28. P. Harremoës, The Information Topology, in: Proceedings IEEE International Symposium on Information Theory, IEEE (2002), p. 431.

    Google Scholar 

  29. P. Harremoës, Information Topologies with Applications (2003), in this volume, pp. 113–150.

    Google Scholar 

  30. P. Harremoës and F. Topsøe, Unified approach to optimization techniques in Shannon theory, in: Proceedings, 2002 IEEE International Symposium on Information Theory, IEEE (2002), p. 238.

    Google Scholar 

  31. P. Harremoës and F. Topsøe, Maximum entropy fundamentals. Entropy, 3 (Sept. 2001), 191–226, http://www.unibas.ch/mdpi/entropy/ [online].

    MathSciNet  Google Scholar 

  32. D. Haussler, A general minimax result for relative entropy, IEEE Trans. Inform. Theory, 43 (1997), 1276–1280.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. S. Holevo, Statistical Structure of Quantum Theory, Springer (Berlin, 2001).

    MATH  Google Scholar 

  34. E. T. Jaynes, Webpage maintained by L. Brethorst, dedicated to Jaynes work, available online from http://bayes.wustl.edu.

    Google Scholar 

  35. E. T. Jaynes, Information theory and statistical mechanics, I and II, Physical Reviews, 106 and 108 (1957), 620–630 and 171–190.

    Article  MathSciNet  Google Scholar 

  36. E. T. Jaynes, Clearing up mysteries — the original goal, in: J Skilling, editor, Maximum Entropy and Bayesian Methods, Kluwer (Dordrecht, 1989).

    Google Scholar 

  37. E. T. Jaynes, Probability Theory — The Logic of Science, Cambridge University Press (Cambridge, 2003).

    MATH  Google Scholar 

  38. A. Jessop, Informed Assessments, an Introduction to Information, Entropy and Statistics, Ellis Horwood (New York, 1995).

    MATH  Google Scholar 

  39. J. N. Kapur, Maximum Entropy Models in Science and Engineering, Wiley (New York, 1993), first edition 1989.

    Google Scholar 

  40. D. Kazakos, Robust noiceless source coding through a game theoretic approach, IEEE Trans. Inform. Theory, 29 (1983), 577–583.

    Article  Google Scholar 

  41. J. L. Kelly, A new interpretation of information rate, Bell System Technical Journal, 35 (1956), 917–926.

    MathSciNet  Google Scholar 

  42. J. Kisynski, Convergence du typé 1, Colloq. Math., 7 (1960), 205–211.

    MATH  MathSciNet  Google Scholar 

  43. S. Kullback, Informaton Theory and Statistics, Wiley (New York, 1959).

    Google Scholar 

  44. S. Kullback and R. Leibler, On information and sufficiency, Ann. Math. Statist., 22 (1951), 79–86.

    MathSciNet  MATH  Google Scholar 

  45. Yu. V. Linnik, An information-theoretic proof of the central limit theorem with Lindeberg condition, Theory Probab. Appl., 4 (1959), 288–299.

    Article  MathSciNet  Google Scholar 

  46. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer (Berlin, Heidelberg, New York, 1993).

    MATH  Google Scholar 

  47. E. Pfaffelhuber, Minimax information gain and minimum discrimination principle, in: I. Csiszár and P. Elias, editors, Topics in Information Theory, volume 16 of Colloquia Mathematica Societatis János Bolyai, János Bolyai Mathematical Society and North-Holland (1977), pp. 493–519.

    Google Scholar 

  48. J. Rissanen, A. Barron and B. Yu, The minimum description length principle in coding and modeling, IEEE Trans. Inform. Theory, 44 (1998), 2743–2760.

    Article  MATH  MathSciNet  Google Scholar 

  49. B. Ya. Ryabko, Comments on “a source matching approach to finding minimax codes”, IEEE Trans. Inform. Theory, 27 (1981), 780–781. Including also the ensuing Editor’s Note.

    Article  MATH  MathSciNet  Google Scholar 

  50. G. Shafer and V. Vovk, Probability and finance. It’s only a game! Wiley (Chichester, 2001).

    Book  Google Scholar 

  51. C. E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J., 27 (1948), 379–423 and 623–656.

    MathSciNet  MATH  Google Scholar 

  52. P. D. Straffin, Game Theory and Strategy, volume 36 of New Mathematical Libary. Mathematical Ass. of America, 1993.

    Google Scholar 

  53. J. J. Sylvester, A question in the geometry of situation, Quarterly Journal of Pure and Applied Mathematics, 1 (1857), 79.

    Google Scholar 

  54. F. Topsøe, An information theoretical identity and a problem involving capacity. Studia Scientiarum Mathematicarum Hungarica, 2 (1967), 291–292.

    MathSciNet  MATH  Google Scholar 

  55. F. Topsøe, A new proof of a result concerning computation of the capacity for a discrete channel, Z. Wahrscheinlichkeitstheorie verw. Geb., 22 (1972), 166–168.

    Article  MATH  Google Scholar 

  56. F. Topsøe, Information theoretical optimization techniques. Kybernetika, 15 (1979), 8–27.

    MathSciNet  Google Scholar 

  57. F. Topsøe, Game theoretical equilibrium, maximum entropy and minimum information discrimination, in: A. Mohammad-Djafari and G. Demoments, editors, Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers (Dordrecht, Boston, London, 1993), pp. 15–23.

    Google Scholar 

  58. F. Topsøe, Basic concepts, identities and inequalities — the toolkit of information theory, Entropy, 3 (2001), 162–190. http://www.unibas.ch/mdpi/entropy/ [online].

    Article  MATH  MathSciNet  Google Scholar 

  59. F. Topsøe, Maximum entropy versus minimum risk and applications to some classical discrete distributions, IEEE Trans. Inform. Theory, 48 (2002), 2368–2376.

    Article  MathSciNet  Google Scholar 

  60. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press (Princeton, 1947), 2nd. edition.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Topsøe, F. (2007). Information Theory at the Service of Science. In: Csiszár, I., Katona, G.O.H., Tardos, G., Wiener, G. (eds) Entropy, Search, Complexity. Bolyai Society Mathematical Studies, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32777-6_8

Download citation

Publish with us

Policies and ethics