On-Line Probability, Complexity and Randomness

  • Authors:

  • Alexey Chernov;Alexander Shen;Nikolai Vereshchagin;Vladimir Vovk

  • Affiliations:

  • Royal Holloway, University of London, Egham, UK TW20 0EX;Marseille and Institute of Information Transmission Problems, LIF (Université Aix-Marseille & CNRS), Moscow;Moscow State University,;Royal Holloway, University of London, Egham, UK TW20 0EX

  • Venue:
  • ALT '08 Proceedings of the 19th international conference on Algorithmic Learning Theory
  • Year:
  • 2008

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Abstract

Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned.We adapt the notions of algorithmic information theory (complexity, algorithmic randomness, martingales, a priori probability) to this framework and show that many classical results are still valid.