On empirical meaning of randomness with respect to a real parameter

TL;DR

This paper explores the concept of randomness in relation to a parameterized family of probability distributions, establishing a link between the computability of the parameter and the measure of random sequences.

Contribution

It demonstrates that the measure of random sequences is positive if and only if the parameter is computable, connecting algorithmic randomness with parameter computability.

Findings

Positive measure of random sequences iff parameter is computable

Discusses methods for generating meaningful random sequences with noncomputable parameters

Links algorithmic randomness to the computability of distribution parameters

Abstract

We study the empirical meaning of randomness with respect to a family of probability distributions $P_\theta$, where $\theta$ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution $P_\theta$ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all $P_\theta$-random sequences is positive if and only if the parameter $\theta$ is a computable real number. The different methods for generating ``meaningful'' $P_\theta$-random sequences with noncomputable $\theta$ are discussed.