Vitanyi, Paul M. B. and Chater, Nick (2017) Identification of probabilities. Journal of Mathematical Psychology, 76 (Part A). pp. 13-24. doi:10.1016/j.jmp.2016.11.004 ISSN 0022-2496.

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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.

Item Type:	Journal Article
Subjects:	B Philosophy. Psychology. Religion > BF Psychology Q Science > QA Mathematics R Medicine > RC Internal medicine > RC0321 Neuroscience. Biological psychiatry. Neuropsychiatry T Technology > TA Engineering (General). Civil engineering (General)
Divisions:	Faculty of Social Sciences > Warwick Business School > Behavioural Science Faculty of Social Sciences > Warwick Business School
Library of Congress Subject Headings (LCSH):	Psychology, Neurosciences, Artificial intelligence, Kolmogorov complexity, Markov processes, Bayesian statistical decision theory
Journal or Publication Title:	Journal of Mathematical Psychology
Publisher:	Elsevier
ISSN:	0022-2496
Official Date:	February 2017
Dates:	Date Event February 2017 Published 26 December 2016 Available 11 November 2016 Accepted
Volume:	76
Number:	Part A
Page Range:	pp. 13-24
DOI:	10.1016/j.jmp.2016.11.004
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	5 December 2016
Date of first compliant Open Access:	3 January 2017
Funder:	European Research Council (ERC), Engineering and Physical Sciences Research Council (EPSRC), Leverhulme Trust (LT), Research Councils UK (RCUK)
Grant number:	295917-RATIONALITY (ERC), ES/K002201/1 (ESRC), RP2012-V-022 (LT), EP/K039830/1 (RCUK)
Open Access Version:	Publisher

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