Trends in Cognitive Sciences
Special Issue: Probabilistic models of cognitionProbabilistic models of cognition: Conceptual foundations
Introduction
The history of probabilistic models of thought is, in a sense, as old as probability theory itself. Probability theory has always had a dual aspect, serving both as a normative theory for ‘correct’ reasoning about chance events, but also as a descriptive theory of how people reason about uncertainty – as providing an analysis, for example, of the mental processes of an ‘intelligent’ juror. The title of Bernouilli's great book, Ars Conjectandi [1], ‘The Art of Conjecture’, nicely embodies this ambiguity, suggesting both a ‘how-to’ guide for better reasoning, and a survey of how the ‘art’ is actually practiced. That is, from its origins, probability theory was viewed as both mathematics and psychology.
From a modern perspective, this conflation seems anomalous. Mathematics has shaken free of its psychological roots, and become an autonomous, and highly formal, discipline. The philosophical thesis of ‘psychologism’, that mathematics (including probability) is a description of thought, fell from favour by the end of the nineteenth century. Moreover, the mathematics and psychology of probability have become divorced. The normative mathematical theory has seen spectacular developments in rigor, generality, and sophistication, going far beyond unaided intuition (see Griffiths and Yuille, Technical Introduction: Supplementary material online). Yet the descriptive study of how people judge probabilities has focussed on apparently systematic patterns of fallacious reasoning about chance [2].
This Special Issue is based on the premise that reconciliation is long overdue and that the mathematics of probability is a vital tool in building theories of cognition. The articles in this issue illustrate how probability provides a rich framework for vision and motor control, learning, language processing, reasoning, and beyond. Moreover, probabilistic models can be applied in various ways – ranging from analyzing a problem that the cognitive system faces, to explicating the function of the specific neural processes that solve it. Rather than advocating a monolithic and exclusively probabilistic view of the mind, we suggest instead that probabilistic methods have a range of valuable roles to play in understanding cognition. We hope that this Special Issue will help further inspire researchers in the cognitive and brain sciences to join the project of illuminating cognition from a probabilistic standpoint; and encourage mathematicians, statisticians and computer scientists to deploy the recent remarkable conceptual and computational armoury that they have developed to help understand cognition.
Section snippets
The ubiquity of probabilistic inference
The cognitive sciences view the brain as an information processor; and information processing typically involves inferring new information from information that has been derived from the senses, from linguistic input, or from memory. This process of inference from old to new is, outside pure mathematics, typically uncertain. Probability theory is, in essence, a calculus for uncertain inference, at least according to the subjective interpretation of probability (Box 1). Thus, prima facie,
Levels of probabilistic explanation
Sophisticated probabilistic models can be related to cognitive processes in a variety of ways. This variety can usefully be understood in terms of Marr's [40] celebrated distinction between three levels of computational explanation: the computational level, which specifies the nature of the cognitive problem being solved, the information involved in solving it, and the logic by which it can be solved; the algorithmic level, which specifies the representations and processes by which solutions to
Conclusion
Sophisticated probabilistic models are finding increasingly wide application across the cognitive and brain sciences. Much of cognition is concerned with dealing, highly effectively, with spectacularly complex problems of probabilistic inference. We suggest that probabilistic methods are likely to be increasingly important theoretical tools for understanding cognition. We hope that the articles in this Special Issue will inspire future researchers to contribute further to the project of
Acknowledgements
This special issue arose from a workshop on ‘Probabilistic Models of Cognition: The Mathematics of Mind’ hosted by the Institute for Pure and Applied Mathematics (IPAM) on the UCLA campus in January 2005. We greatly thank IPAM, in particular the director Mark Green and the advisory board, for their leadership role in recognizing early on the scientific potential of this emerging area of mathematical modeling. IPAM, and its enthusiastic staff, provided intellectual and financial support,
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