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Algorithmic Identification of Probabilities Is Hard

  • Laurent Bienvenu
  • Benoît Monin
  • Alexander Shen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8776)

Abstract

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter p. By the law of large numbers, the frequency of zeros in the sequence tends to p, and thus we can get better and better approximations of p as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that p is a computable real, and one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter p (in the form of its Turing code). Can one do such a thing uniformly for all sequences that are random for computable Bernoulli measures, or even for a ‘large enough’ fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures.

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References

  1. [Gác05]
    Gács, P.: Uniform test of algorithmic randomness over a general space. Theoretical Computer Science 341(1-3), 91–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [LV08]
    Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its applications, 3rd edn. Texts in Computer Science. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  3. [VC13]
    Vitanyi, P., Chater, N.: Algorithmic identification of probabilities (2013), http://arxiv.org/abs/1311.7385
  4. [Wei00]
    Weihrauch, K.: Computable analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  5. [ZZ08]
    Zeugmann, T., Zilles, S.: Learning recursive functions: a survey. Theoretical Computer Science 397, 4–56 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Laurent Bienvenu
    • 1
  • Benoît Monin
    • 2
  • Alexander Shen
    • 3
  1. 1.Laboratoire PonceletFrance
  2. 2.LIAFAFrance
  3. 3.LIRMM (on leave from IITP RAS)France

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