(Non-)Equivalence of Universal Priors

  • Ian Wood
  • Peter Sunehag
  • Marcus Hutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7070)

Abstract

Ray Solomonoff invented the notion of universal induction featuring an aptly termed “universal” prior probability function over all possible computable environments [9]. The essential property of this prior was its ability to dominate all other such priors. Later, Levin introduced another construction — a mixture of all possible priors or “universal mixture”[12]. These priors are well known to be equivalent up to multiplicative constants. Here, we seek to clarify further the relationships between these three characterisations of a universal prior (Solomonoff’s, universal mixtures, and universally dominant priors). We see that the the constructions of Solomonoff and Levin define an identical class of priors, while the class of universally dominant priors is strictly larger. We provide some characterisation of the discrepancy.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chaitin, G.J.: A theory of program size formally identical to information theory. Journal of the ACM 22, 329–340 (1975)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Downey, R.G., Hirschfeldt, D.R.: Kolmogorov complexity of finite strings. In: Algorithmic Randomness and Complexity, pp. 110–153. Springer, New York (2010)CrossRefGoogle Scholar
  3. 3.
    Figueira, S., Stephan, F., Wu, G.: Randomness and universal machines. Journal of Complexity 22(6), 738–751 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hutter, M.: Universal artificial intelligence: Sequential decisions based on algorithmic probability. Springer (2005)Google Scholar
  5. 5.
    Hutter, M.: On universal prediction and Bayesian confirmation. Theoretical Computer Science 384(1), 33–48 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Levin, L.A.: Some Theorems on the Algorithmic Approach to Probability Theory and Information Theory. Phd dissertation, Moscow University, Moscow (1971)Google Scholar
  7. 7.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer (2008)Google Scholar
  8. 8.
    Schnorr, C.: Process complexity and effective random tests. Journal of Computer and System Sciences 7(4), 376–388 (1973)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Solomonoff, R.J.: A formal theory of inductive inference. parts I and II. Information and Control 7(2), 224–254 (1964)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Solomonoff, R.J.: Complexity-based induction systems: Comparisons and convergence theorems. IEEE Transactions on Information Theory 24(4), 422–432 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42(2), 230–265 (1936)MathSciNetGoogle Scholar
  12. 12.
    Zvonkin, A.K., Levin, L.A.: 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), 83–124 (1970)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ian Wood
    • 1
  • Peter Sunehag
    • 1
  • Marcus Hutter
    • 1
    • 2
  1. 1.School of Computer ScienceThe Australian National UniversityCanberraAustralia
  2. 2.Department of Computer ScienceETH ZürichSwitzerland

Personalised recommendations