Conditional Probabilities and van Lambalgen’s Theorem Revisited

Bruno Bauwens (1) , Alexander Shen (2) , Hayato Takahashi (3)

1 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
2 ESCAPE - Systèmes complexes, automates et pavages
3 Organization for Promotion of Higher Education and Student Support

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the 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., [8] or [7] for a short introduction).

Mots clés: Van Lambalben theorem; Conditional probability

Domaines: Informatique [cs]; Mathématiques [math]

Bruno Bauwens, Alexander Shen, Hayato Takahashi. Conditional Probabilities and van Lambalgen’s Theorem Revisited. Theory of Computing Systems, 2017, 61 (4), pp.1315-1336. ⟨10.1007/s00224-017-9789-2⟩. ⟨lirmm-01486477⟩