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Algorithmic learning theory traditionally studies the learnability ofeffective infinite binary sequences (reals), while recent work by [Vitanyi andChater, 2017] and [Bienvenu et al., 2014] has adapted this framework to thestudy of learnability of effective probability distributions from random data.We prove that for certain families of probability measures that areparametrized by reals, learnability of a subclass of probability measures isequivalent to learnability of the class of the corresponding real parameters.This equivalence allows to transfer results from classical algorithmic theoryto learning theory of probability measures. We present a number of suchapplications, providing many new results regarding EX and BC learnability ofclasses of measures, thus drawing parallels between the two learning theories.

Equivalences between learning of data and probability distributions, and their applications