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

Abstract

We study the a priori semimeasure of sets of P θ -random infinite sequences, where P θ is a family of probability distributions depending on a real parameter θ. In the case when for a computable probability distribution P θ an effectively strictly consistent estimator exists, we show that Levin’s a priory semimeasure of the set of all P θ -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 $\bigcup_{\theta}I_{\theta}$, where I θ is the set of all P θ -random sequences and the union is taken over all algorithmically non-random θ, is positive.