A quantum theoretical explanation for probability judgment errors.

Abstract

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Quantum Probability 1 Permanent repository link:Abstract A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction and disjunction fallacies, averaging effects, unpacking effects, and order effects on inference. On the one hand, quantum theory is similar to other categorization and memory models of cognition in that it relies on vector spaces defined by features, and similarities between vectors to determine probability judgments. On the other hand, quantum probability theory is a generalization of Bayesian probability theory because it is based on a set of (von Neumann) axioms that relax some of the classic (Kolmogorov) axioms. The quantum model is compared and contrasted with other competing explanations for these judgment errors including the anchoring and adjustment model for probability judgments. The quantum model introduces a new fundamental concept to cognition --the compatibility versus incompatibility of questions and the effect this can have on the sequential order of judgments.We conclude that quantum information processing principles provide a viable and promising new way to understand human judgment and reasoning. Quantum Probability 2Over 30 years ago, Kahneman and Tversky (1982) began their influential program of research to discover the heuristics and biases that form the basis of human probability judgments.Since that time, a great deal of new and challenging empirical phenomena have been discovered including conjunction and disjunction fallacies, unpacking effects, and order effects on inference (Gilovich, Griffin, & Kahneman, 2002). Although heuristic concepts (such as representativeness, availability, anchor-adjustment) initially served as a guide to researchers in this area, there is a growing need to move beyond these intuitions, and develop more coherent, comprehensive, and deductive theoretical explanations (Shah & Oppenheimer, 2008). The purpose of this article is to propose a new way of understanding human probability judgment using quantum probability principles (Gudder, 1988).At first, it might seem odd to apply quantum theory to human judgments. Before we address this general issue, we point out that we are not claiming the brain to be a quantum computer; rather we only use quantum principles to derive cognitive models and leave the neural basis for later research. That is, we use the mathematical principles of quantum probability detached from the physical meaning associated with quantum mechanics. This approach is similar to the application of complexity theory or stochastic processes to domains outside of physics. 1There are at least five reasons for doing so: (1) judgment is not a simple read out from a pre-existing or recorded state, instead it is constructed from the question and the cognitive state created by the current context; from this first point it then follows that (2) drawing a conclus...

The conjunction fallacy, confirmation, and quantum theory: Comment on Tentori, Crupi, and Russo (2013).

“…The latter is based on the intuition that, because almost all women own black shoes, the Linda story does not produce any increase in the likelihood of owning black shoes, so the right hand side is zero. Now consider a simple QP model for this case (for a general introduction to QP theory, see Busemeyer et al 2011 andBruza 2012). The reader will notice that we have to make more assumptions than the IC hypothesis to account for findings presented in TCR.…”

“…That is, the same principles are applied uniformly across all four examples. Furthermore, these same principles are used to account for many other phenomena, not covered by the IC hypothesis, such as conjunction fallacies with more than two events, disjunction fallacies, unpacking effects, and order effects on inference (Busemeyer et al, 2011).In general, a person's state of beliefs about the presence or absence of various feature combinations is represented by a (unit length) vector in an N −dimensional space. For simplicity, we limit the following applications to a 4−dimensional space.…”

A quantum probability framework for human probabilistic inference.

“…It is generally believed that experience with a particular situation, either from previous familiarity or acquired through learning, may allow events to be represented in a compatible way, whereas relatively novel situations are more likely to be represented in a incompatible way. In addition, quantum models are often used to explain similar phenomena as heuristics (Busemeyer et al, 2011), and so it seems plausible that incompatible representations of A Quantum Framework for Probabilistic Inference 7 events, associated with quantum models, should be preferentially used for decisions executed spontaneously with little conscious deliberation.Compatible events are ones that may be assigned a simultaneous truth value. Thus, if event X and event Y are compatible, their conjunction X ∧Y is well defined.…”

The rational status of quantum cognition.

“…An illustration of how the CE emerges in a QPT approach to Tversky and Kahneman's (1983) Linda paradigm. The two-dimensional representation is a caricature and the QPT model proper must be specified with subspaces of arbitrary dimensionality (Busemeyer et al, 2011). Figure 1b.…”

“…Determining whether two questions are incompatible in the first place is a challenge to the QPT CE approach and we summarize the progress to date (Busemeyer et al, 2011(Busemeyer et al, , 2015. A marker of incompatibility would be order effects, that is the extent to which assessing X and then Y produces the same responses as assessing Y and then X.…”