How much randomness is needed for statistics?

TL;DR

This paper investigates the minimal amount of information about a measure needed for defining randomness, comparing classical and Hippocratic approaches across different randomness notions.

Contribution

It extends previous results by showing that classical and Hippocratic randomness notions coincide for Bernoulli measures only in Martin-Löf randomness, not in computable randomness or stochasticity.

Findings

Classical and Hippocratic randomness coincide for Martin-Löf randomness with Bernoulli measures.

The equivalence does not hold for computable randomness and stochasticity.

The paper clarifies the limitations of the Hippocratic approach in various randomness contexts.

Abstract

In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $\lambda $, a choice needs to be made. One approach is to allow randomness tests to access the measure $\lambda $ as an oracle (which we call the "classical approach"). The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in $\lambda $ (we call this approach "Hippocratic"). While the Hippocratic approach is in general much more restrictive, there are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-L\"of randomness and the measure $\lambda $ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.