On Martin-Löf Convergence of Solomonoff’s Mixture

  • Tor Lattimore
  • Marcus Hutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7876)


We study the convergence of Solomonoff’s universal mixture on individual Martin-Löf random sequences. A new result is presented extending the work of Hutter and Muchnik (2004) by showing that there does not exist a universal mixture that converges on all Martin-Löf random sequences.


Solomonoff induction Kolmogorov complexity theory of computation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Calude, C.: Information and Randomness: An Algorithmic Perspective, 2nd edn. Springer-Verlag New York, Inc., Secaucus (2002)CrossRefMATHGoogle Scholar
  2. 2.
    Hutter, M.: On universal prediction and Bayesian confirmation. Theoretical Computer Science 384(1), 33–48 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hutter, M., Muchnik, A.: Universal convergence of semimeasures on individual random sequences. In: Ben-David, S., Case, J., Maruoka, A. (eds.) ALT 2004. LNCS (LNAI), vol. 3244, pp. 234–248. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Hutter, M., Muchnik, A.: On semimeasures predicting Martin-Löf random sequences. Theoretical Computer Science 382(3), 247–261 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Li, M., Vitanyi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer (2008)Google Scholar
  6. 6.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9(6), 602–619 (1966)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Miyabe, K.: An optimal superfarthingale and its convergence over a computable topological space. In: Solomonoff Memorial. LNCS. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Rathmanner, S., Hutter, M.: A philosophical treatise of universal induction. Entropy 13(6), 1076–1136 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Solomonoff, R.: A formal theory of inductive inference, Part I. Information and Control 7(1), 1–22 (1964)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Solomonoff, R.: Complexity-based induction systems: Comparisons and convergence theorems. IEEE Transactions on Information Theory 24(4), 422–432 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Vovk, V.: On a randomness criterion. Soviet Mathematics Doklady 35, 656–660 (1987)MATHGoogle Scholar
  12. 12.
    Willems, F., Shtarkov, Y., Tjalkens, T.: The context tree weighting method: Basic properties. IEEE Transactions on Information Theory 41, 653–664 (1995)CrossRefMATHGoogle Scholar
  13. 13.
    Wood, I., Sunehag, P., Hutter, M. (Non-)equivalence of universal priors. In: Solomonoff Memorial. LNCS. Springer, Heidelberg (2011)Google Scholar
  14. 14.
    Zvonkin, A., Levin, L.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25(6), 83 (1970)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tor Lattimore
    • 1
  • Marcus Hutter
    • 1
  1. 1.Australian National UniversityAustralia

Personalised recommendations