Theory of Computing Systems

, Volume 61, Issue 4, pp 1315–1336 | Cite as

Conditional Probabilities and van Lambalgen’s Theorem Revisited

Article
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)

Abstract

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).

Keywords

Conditional probability Algorithmic randomness van Lambalgen’s theorem 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.National Research University Higher School of Economics (HSE)Faculty of Computer ScienceMoscowRussia
  2. 2.Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier, CNRSUniversité de MontpellierMontpellierFrance
  3. 3.Organization for Promotion of Higher Education and Student SupportGifu UniversityGifu CityJapan
  4. 4.Random Data LaboratoryTokyoJapan

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