On Empirical Meaning of Randomness with Respect to Parametric Families of Probability Distributions

Vladimir V'yugin

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We study the a priori semimeasure of sets of Pθ-random infinite sequences, where Pθ is a family of probability distributions depending on a real parameter θ. In the case when for a computable probability distribution Pθ an effectively strictly consistent estimator exists, we show that Levin's a priory semimeasure of the set of all Pθ-random sequences is positive if and only if the parameter θ is a computable real number. We show that the a priory semimeasure of the set∪θ, where Iθ is the set of all Pθ-random sequences and the union is taken over all algorithmically non-random θ, is positive.

Original languageEnglish
Pages (from-to)296-312
Number of pages17
JournalTheory of Computing Systems
Volume50
Issue number2
DOIs
Publication statusPublished - Feb 2012
Externally publishedYes

Keywords

  • A priory semimeasure
  • Algorithmic information theory
  • Bernoully sequences
  • Martin-Löf random sequences
  • Parametric families of probability distributions
  • Probabilistic machines
  • Turing degrees

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