On Empirical Meaning of Randomness with Respect to Parametric Families of Probability Distributions
We study the a priori semimeasure of sets of P(theta)-random infinite sequences, where P(theta) is a family of probability distributions depending on a real parameter theta. In the case when for a computable probability distribution P(theta) an effectively strictly consistent estimator exists, we show that Levin's a priory semimeasure of the set of all P(theta)-random sequences is positive if and only if the parameter theta is a computable real number. We show that the a priory semimeasure of the set U(theta)I(theta), where I(theta) is the set of all P(theta)-random sequences and the union is taken over all algorithmically non-random theta, is positive.
parametric families of probability distributions a priory semimeasure martin-lof random sequences probabilistic machines bernoully sequences algorithmic information theory turing degrees