Synthese

, Volume 53, Issue 3, pp 419–432 | Cite as

The significance of the ergodic decomposition of stationary measures for the interpretation of probability

  • Jan Von Plato
The Foundations Of Probability

Abstract

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

Keywords

Dynamical System Probability Measure Stationary Measure Stationary Probability Ergodic Theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Billingsley, P.: Ergodic Theory and Information, Wiley, New York, 1965.Google Scholar
  2. Choquet, G.: Lectures on Analysis, vol. 2, Benjamin, New York, 1969.Google Scholar
  3. Diaconis, P. and D. Freedman: ‘de Finetti's generalizations of exchangeability’, in R. Jeffrey (ed.), pp. 233–249.Google Scholar
  4. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed., Wiley, New York, 1971.Google Scholar
  5. de Finetti, B.: ‘La prévision: ses lois logiques, ses sources subjectives’, Ann. Inst. Henri Poincaré 7 (1937), pp. 1–68. English translation in Kyburg and Smokler (eds.) Studies in Subjective Probability, Wiley, New York, 1964.Google Scholar
  6. de Finetti, B.: ‘Sur la condition d'équivalence partielle’, in Colloque consacré a la theorie des probabilités, vol. VI, pp. 5–18. (= Act. Sci. Ind 739), Hermann, Paris, 1938. English translation in R. Jeffrey (ed.), pp. 193–205.Google Scholar
  7. de Finetti, B.: ‘Initial probabilities: a prerequisite for any valid induction’, Synthese 20 (1969), 2–16.Google Scholar
  8. de Finetti, B.: Probability, Induction and Statistics, wiley, New York, 1972.Google Scholar
  9. de Finetti, B.: Theory of Probability, vol. 1, Wiley, New York, 1974.Google Scholar
  10. Freedman, D.: ‘Invariants under mixing which generalize de Finetti's theorem’, Ann. Math. Stat. 33 (1962), 916–923.Google Scholar
  11. Good, I. J.: The Estimation of Probabilities. M.I.T. Press, Cambridge, Mass., 1965.Google Scholar
  12. Grenander, U. and M. Rosenblatt: Statistical Analysis of Stationary Time Series, Wiley, New York, 1962.Google Scholar
  13. Hintikka, J.: ‘Unknown probabilities, Bayesianism and de Finetti's representation theorem’, in R. C. Buck and R. S. Cohen (eds.), PSA 1970, Boston Studies in the Philosophy of Science, vol. VIII, pp. 325–341, Reidel, Dordrecht, 1971.Google Scholar
  14. Hopf, E.: ‘On causality, statistics and probability’, Journal of Mathematics and Physics 17 (1934), 51–102.Google Scholar
  15. Jacobs, K.: Lecture Notes on Ergodic Theory, Aarhus Universitet, Matematisk Institut, Aarhus, 1963 (mimeographed).Google Scholar
  16. Jeffrey, R.: (ed.) Studies in Inductive Logic and Probability, vol. 2, University of California Press, Berkeley, 1980.Google Scholar
  17. Khinchin, A. I.: Mathematical Foundations of Statistical Mechanics, Dover, New York, 1949.Google Scholar
  18. Kriloff, N. and N. Bogoliouboff: ‘La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non-linéaire’, Annals of Mathematics, 2nd ser. 38 (1937), 65–113.Google Scholar
  19. Link, G.: ‘Representation theorems of the de Finetti type for (partially) symmetric probability measures’, in R. Jeffrey (ed.), pp. 207–231.Google Scholar
  20. Malament, D. and S. Zabell: ‘Why Gibbs averages work — the role of ergodic theory’, Philosophy of Science 47 (1980) 339–349.Google Scholar
  21. Martin-Löf, A.: Statistical Mechanics and the Foundations of Thermodynamics, Lecture Notes in Physics, vol. 101, Springer, Berlin, 1979.Google Scholar
  22. von Neumann, J.: ‘Zur Operatorenmethode in der klassischen Mechanik’, Annals of Mathematics 33 (1932), 587–642. Reprinted in von Neumann, Collected Works, vol. II.Google Scholar
  23. Ornstein, D.: Ergodic Theory, Randomness, and Dynamical Systems, Yale University Press, New Haven and London, 1974.Google Scholar
  24. Phelps, R.: Lectures on Choquet's Theorem, Van Nostrand, New York, 1966.Google Scholar
  25. von Plato, J.: ‘Epistemological problems of subjective probabilities’, Reports from the Department of Philosophy, University of Helsinki, No 2, 1980.Google Scholar
  26. von Plato, J.: ‘Reductive relations in interpretations of probability’, Synthese 48 (1981), 61–75.Google Scholar
  27. Ruelle, D.: Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.Google Scholar
  28. Ryll-Nardzewski, C.: ‘On stationary sequences of random variables and the de Finetti's equivalence’, Colloquium Mathematicum 4 (1957), 146–156.Google Scholar
  29. Stegmüller, W.: Personelle und Statistische Wahrscheinlichkeit, 2 vols. Springer, Berlin, 1973.Google Scholar
  30. Truesdell, C.: ‘Ergodic theory in classical statistical mechanics’, in Caldirola (ed.), Ergodic Theories, pp. 21–56, Academic Press, New York and London, 1961.Google Scholar

Copyright information

© D. Reidel Publishing Co 1982

Authors and Affiliations

  • Jan Von Plato
    • 1
  1. 1.University of HelsinkiFinland

Personalised recommendations