A note on the use of probabilities by mechanical learners

Abstract

We raise the following problem: in a probabilistic context, is it always fruitful for a machine to compute probabilities? The question is made precise in a paradigm of the limit-identification kind, where a learner must discover almost surely whether an infinite sequence of heads and tails belongs to an effective subset S of the Cantor space. In this context, a successful strategy for an ineffective learner is to compute, at each stage, the conditional probability that he is faced with an element of S, given the data received so far. We show that an effective learner should not proceed this way in all circumstances. Indeed, even if he gets an optimal description of a set S, and even if some machine can always compute the conditional probability for S given any data, an effective learner optimizes his inductive competence only if he does not compute the relevant probabilities. We conclude that the advice “compute probabilities whenever you can” should sometimes be received with caution in the context of machine learning.

We thank Scott Weinstein, Michiel van Lambalgen, and several anonymous reviewers for helpful comments. Research support was provided by the Office of Naval Research under contracts Nos. N00014-87-K-0401 N00014-89-J-1725, by the Swiss National Science Foundation under grant number 21-32399.91, and by the Centre National de la Recherche Scientifique.