Theory of Computing Systems

, Volume 61, Issue 4, pp 1315–1336 | Cite as

Conditional Probabilities and van Lambalgen’s Theorem Revisited

  • Bruno Bauwens
  • Alexander Shen
  • Hayato Takahashi
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)


The definition of conditional probability in the case of continuous distributions (for almost all conditions) was an important step in the development of mathematical theory of probabilities. Can we define this notion in algorithmic probability theory for individual random conditions? Can we define randomness with respect to the conditional probability distributions? Can van Lambalgen’s theorem (relating randomness of a pair and its elements) be generalized to conditional probabilities? We discuss the developments in this direction. We present almost no new results trying to put known results into perspective and explain their proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness (see, e.g., Shen et al. ??2013 or Shen ??2015 for a short introduction).


Conditional probability Algorithmic randomness van Lambalgen’s theorem 



We are grateful to the organizers of the “Focus Semester on Algorithmic Randomness” (June 2015): Klaus Ambos-Spies, Anja Kamp, Nadine Losert, Wolfgang Merkle, and Martin Monath. We thank the Heidelberg university and Templeton foundation for financial support. The visit of Hayato Takahashi to LIRMM was supported by NAFIT ANR-08-EMER-008-01 grant.

Alexander Shen thanks Vitaly Arzumanyan, Alexey Chernov, Andrei Romashchenko, Nikolay Vereshchagin, and all members of Kolmogorov seminar group in Moscow and ESCAPE team in Montpellier.

Hayato Takahashi was supported by JSPS KAKENHI grant number 24540153.

Alexander Shen was supported by ANR-15-CE40-0016-01 RaCAF grant.

Last but not least, we are grateful to anonymous referees for very detailed reviews and many corrections and suggestions.


  1. 1.
    Ackerman, N.L., Freer, C.E., Roy, D.M.: Noncomputable conditional distributions 26th Annual IEEE Symposium on Logic in Computer Science (LICS), pp 107–116 (2011). See also: On the computability of conditional probability, arXiv:1005.3014v2.pdf Google Scholar
  2. 2.
    Bauwens, B.: Conditional measure and the violation of van Lambalgen’s theorem for Martin-Löf randomness. arXiv:1509.02884 (2015)
  3. 3.
    Bienvenu, L., Gács, P., Hoyrup, M., Rojas, C., Shen, A.: Algorithmic tests and randomness with respect to a class of measures Proceedings of the Steklov Institute of Mathematics, vol. 274, pp 34–89 (2011). doi: 10.1134/S0081543811060058. See also arXiv:1103.1529 Google Scholar
  4. 4.
    Kjos-Hannsen, B.: The probability distribution as a computational resource for randomness testing. J Log Anal 2(1), 1–13 (2010). doi: 10.4115/jla.2010.2.10. See also arXiv:1408:2850 MathSciNetGoogle Scholar
  5. 5.
    de Leeuw, K., Moore, E.F., Shannon, C.E., Shapiro, N.: Computability by probabilistic machines. Automata studies, edited by C.E. Shannon and J. McCarthy, Annals of Mathematics studies no. 34, lithoprinted, pp 183–212. Princeton University Press, Princeton (1956)Google Scholar
  6. 6.
    van Lambalgen, M.: The axiomatization of randomness. J. Symb. Log. 55(3), 1143–1167 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Shen, A.: Around Kolmogorov complexity: basic notions and results Measures of Complexity: Festschrift for Alexey Chervonenkis. See also: arXiv:1504.04955, pp 75–116. Springer (2015)
  8. 8.
    Shen, A., Uspensky, V., Vereshchagin, N.: Kolmogorov complexity and algorithmic randomness, Moscow, MCCME. English version: (2013)
  9. 9.
    Takahashi, H.: On a definition of random sequences with respect to conditional probability. Inf. Comput. 206(12), 1375–1382 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Takahashi, H.: Algorithmic randomness and monotone complexity on product space. Inf. Comput. 209(2), 183–197 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Takahashi, H.: Generalization of van Lambalgen’s theorem and blind randomness for conditional probability, arXiv:1310.0709
  12. 12.
    Vovk, V.G., V’yugin, V.V.: On the empirical validity of the Bayesian method. J. R. Stat. Soc. Ser. B Methodol. 55(1), 253–266 (1993)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.National Research University Higher School of Economics (HSE)Faculty of Computer ScienceMoscowRussia
  2. 2.Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier, CNRSUniversité de MontpellierMontpellierFrance
  3. 3.Organization for Promotion of Higher Education and Student SupportGifu UniversityGifu CityJapan
  4. 4.Random Data LaboratoryTokyoJapan

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