Abstract
We study the empirical meaning of randomness with respect to a family of probability distributions P θ , where θ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution P θ an effectively strongly consistent estimate exists, we show that the Levin’s a priory semicomputable semimeasure of the set of all P θ -random sequences is positive if and only if the parameter θ is a computable real number. The different methods for generating “meaningful” P θ -random sequences with noncomputable θ are discussed.
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V’yugin, V. (2007). On Empirical Meaning of Randomness with Respect to a Real Parameter. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds) Computer Science – Theory and Applications. CSR 2007. Lecture Notes in Computer Science, vol 4649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74510-5_39
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DOI: https://doi.org/10.1007/978-3-540-74510-5_39
Publisher Name: Springer, Berlin, Heidelberg
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