Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
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We study the empirical meaning of randomness with respect to a family of probability distributions Pθ, where θ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution Pθ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all Pθ-random sequences is positive if and only if the parameter θ is a computable real number. The different methods for generating "meaningful" Pθ-random sequences with noncomputable θ are discussed.