title: Identification of probabilities.
abstract: Within psychology, neuroscience and artificial intelligence, there has been increasing interest in the proposal that the brain builds probabilistic models of sensory and linguistic input: that is, to infer a probabilistic model from a sample. The practical problems of such inference are substantial: the brain has limited data and restricted computational resources. But there is a more fundamental question: is the problem of inferring a probabilistic model from a sample possible even in principle? We explore this question and find some surprisingly positive and general results. First, for a broad class of probability distributions characterized by computability restrictions, we specify a learning algorithm that will almost surely identify a probability distribution in the limit given a finite i.i.d. sample of sufficient but unknown length. This is similarly shown to hold for sequences generated by a broad class of Markov chains, subject to computability assumptions. The technical tool is the strong law of large numbers. Second, for a large class of dependent sequences, we specify an algorithm which identifies in the limit a computable measure for which the sequence is typical, in the sense of Martin-Löf (there may be more than one such measure). The technical tool is the theory of Kolmogorov complexity. We analyze the associated predictions in both cases. We also briefly consider special cases, including language learning, and wider theoretical implications for psychology.
authors.affiliation: National Research Institute for Mathematics and Computer Science, CWI, Science Park 123, 1098 XG, Amsterdam, The Netherlands; University of Amsterdam, The Netherlands.
authors.forename: Paul M B
authors.initials: PM
authors.lastname: Vitányi
authors.affiliation: Behavioural Science Group, Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK.
authors.forename: Nick
authors.initials: N
authors.lastname: Chater
keywords: Bayesian brain, identification; Computable measure; Computable probability; Kolmogorov complexity; Learning; Markov chain; Martin-Löf randomness; Strong law of large numbers; Typicality