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      Bayesian reasoning with ifs and ands and ors

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          Abstract

          The Bayesian approach to the psychology of reasoning generalizes binary logic, extending the binary concept of consistency to that of coherence, and allowing the study of deductive reasoning from uncertain premises. Studies in judgment and decision making have found that people’s probability judgments can fail to be coherent. We investigated people’s coherence further for judgments about conjunctions, disjunctions and conditionals, and asked whether their coherence would increase when they were given the explicit task of drawing inferences. Participants gave confidence judgments about a list of separate statements (the statements group) or the statements grouped as explicit inferences (the inferences group). Their responses were generally coherent at above chance levels for all the inferences investigated, regardless of the presence of an explicit inference task. An exception was that they were incoherent in the context known to cause the conjunction fallacy, and remained so even when they were given an explicit inference. The participants were coherent under the assumption that they interpreted the natural language conditional as it is represented in Bayesian accounts of conditional reasoning, but they were incoherent under the assumption that they interpreted the natural language conditional as the material conditional of elementary binary logic. Our results provide further support for the descriptive adequacy of Bayesian reasoning principles in the study of deduction under uncertainty.

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          Decision by sampling.

          We present a theory of decision by sampling (DbS) in which, in contrast with traditional models, there are no underlying psychoeconomic scales. Instead, we assume that an attribute's subjective value is constructed from a series of binary, ordinal comparisons to a sample of attribute values drawn from memory and is its rank within the sample. We assume that the sample reflects both the immediate distribution of attribute values from the current decision's context and also the background, real-world distribution of attribute values. DbS accounts for concave utility functions; losses looming larger than gains; hyperbolic temporal discounting; and the overestimation of small probabilities and the underestimation of large probabilities.
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            Optimal predictions in everyday cognition.

            Human perception and memory are often explained as optimal statistical inferences that are informed by accurate prior probabilities. In contrast, cognitive judgments are usually viewed as following error-prone heuristics that are insensitive to priors. We examined the optimality of human cognition in a more realistic context than typical laboratory studies, asking people to make predictions about the duration or extent of everyday phenomena such as human life spans and the box-office take of movies. Our results suggest that everyday cognitive judgments follow the same optimal statistical principles as perception and memory, and reveal a close correspondence between people's implicit probabilistic models and the statistics of the world.
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              Can quantum probability provide a new direction for cognitive modeling?

              Classical (Bayesian) probability (CP) theory has led to an influential research tradition for modeling cognitive processes. Cognitive scientists have been trained to work with CP principles for so long that it is hard even to imagine alternative ways to formalize probabilities. However, in physics, quantum probability (QP) theory has been the dominant probabilistic approach for nearly 100 years. Could QP theory provide us with any advantages in cognitive modeling as well? Note first that both CP and QP theory share the fundamental assumption that it is possible to model cognition on the basis of formal, probabilistic principles. But why consider a QP approach? The answers are that (1) there are many well-established empirical findings (e.g., from the influential Tversky, Kahneman research tradition) that are hard to reconcile with CP principles; and (2) these same findings have natural and straightforward explanations with quantum principles. In QP theory, probabilistic assessment is often strongly context- and order-dependent, individual states can be superposition states (that are impossible to associate with specific values), and composite systems can be entangled (they cannot be decomposed into their subsystems). All these characteristics appear perplexing from a classical perspective. However, our thesis is that they provide a more accurate and powerful account of certain cognitive processes. We first introduce QP theory and illustrate its application with psychological examples. We then review empirical findings that motivate the use of quantum theory in cognitive theory, but also discuss ways in which QP and CP theories converge. Finally, we consider the implications of a QP theory approach to cognition for human rationality.
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                Author and article information

                Contributors
                Journal
                Front Psychol
                Front Psychol
                Front. Psychol.
                Frontiers in Psychology
                Frontiers Media S.A.
                1664-1078
                25 February 2015
                2015
                : 6
                : 192
                Affiliations
                [1] 1Department of Psychological Sciences, Birkbeck, University of London , London, UK
                [2] 2Laboratory CHArt (PARIS), Université Paris 8 , Paris, France
                [3] 3Institut Jean Nicod , Paris, France
                [4] 4Department of Psychology, Durham University , Durham, UK
                Author notes

                Edited by: David R. Mandel, Defence Research and Development Canada, Toronto Research Centre, Canada

                Reviewed by: Niki Pfeifer, Ludwig-Maximilians-Universität München, Germany; Igor Douven, University of Groningen, Netherlands

                *Correspondence: David E. Over, Department of Psychology, Durham University, Durham University Science Site, South Road, Durham DH1 3LE, UK e-mail: david.over@ 123456durham.ac.uk

                This article was submitted to Cognition, a section of the journal Frontiers in Psychology.

                Article
                10.3389/fpsyg.2015.00192
                4340177
                25762965
                2a42dfea-c0b7-4f90-b177-e59936b82e0b
                Copyright © 2015 Cruz, Baratgin, Oaksford and Over.

                This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

                History
                : 14 December 2014
                : 06 February 2015
                Page count
                Figures: 2, Tables: 1, Equations: 11, References: 61, Pages: 9, Words: 8973
                Categories
                Psychology
                Original Research Article

                Clinical Psychology & Psychiatry
                uncertain reasoning,deduction,conditionals,coherence,conjunction fallacy

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