Conditional Probabilities and van Lambalgen’s Theorem Revisited
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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).
KeywordsConditional 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.
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