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Cognitive Science

A Quantum Probability-theoretic account of human judgment using Positive-Operator-Valued Measures

Takayuki Miyadera, Steven D. Phillips

Published Jan 1, 2012• 1 citation
OpenAlex

Abstract

People make logically inconsistent probability judgments.The "Linda" problem is a well-known example, which often elicits a conjunction/disjunction fallacy: probability of constituent event A (B) judged more/less likely than their conjunction/disjunction.The Quantum Judgment model (QJM, Busemeyer et al 2011) explains such errors, which are not explainable within classical probability theory.We propose an alternative axiomatic approach in the framework of quantum probability theory that employs positive operators representing the set of general queries, in constast to QJM which uses projection operators.Like QJM, our model accounts for conjunction/disjunction fallacies, averaging type errors, and unpacking effects, suggesting that it provides a viable model of judgement error.Further differences between our model and QJM are also discussed.

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People make logically inconsistent probability judgments.The "Linda" problem is a well-known example, which often elicits a conjunction/disjunction fallacy: probability of constituent event A (B) judged more/less likely than their conjunction/disjunction.The Quantum Judgment model (QJM, Busemeyer et al 2011) explains such errors, which are not explainable within classical probability theory.We propose an alternative axiomatic approach in the framework of quantum probability theory that employs positive operators representing the set of general queries, in constast to QJM which uses projection operators.Like QJM, our model accounts for conjunction/disjunction fallacies, averaging type errors, and unpacking effects, suggesting that it provides a viable model of judgement error.Further differences between our model and QJM are also discussed.
A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction, disjunction, inverse, and conditional fallacies, as well as unpacking effects and partitioning effects. Quantum probability theory is a general and coherent theory based on a set of (von Neumann) axioms which relax some of the constraints underlying classic (Kolmogorov) probability theory. The quantum model is compared and contrasted with other competing explanations for these judgment errors including the representativeness heuristic, the averaging model, and a memory retrieval model for probability judgments. The quantum model also provides ways to extend Bayesian, fuzzy set, and fuzzy trace theories. We conclude that quantum information processing principles provide a viable and promising new way to understand human judgment and reasoning.
How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B , which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.
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. In the quantum model, a new fundamental concept in cognition is advanced--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.
People can often outperform statistical methods and machine learning algorithms in situations that involve making inferences about the relationship between causes and effects. While people are remarkably good at causal reasoning in many situations, there are several instances where they deviate from expected responses. This paper examines three situations where judgments related to causal inference problems produce unexpected results and describes a quantum inference model based on the axiomatic principles of quantum probability theory that can explain these effects. Two of the three phenomena arise from the comparison of predictive judgments (i.e., the conditional probability of an effect given a cause) with diagnostic judgments (i.e., the conditional probability of a cause given an effect). The third phenomenon is a new finding examining order effects in predictive causal judgments. The quantum inference model uses the notion of incompatibility among different causes to account for all three phenomena. Psychologically, the model assumes that individuals adopt different points of view when thinking about different causes. The model provides good fits to the data and offers a coherent account for all three causal reasoning effects thus proving to be a viable new candidate for modeling human judgment.
The standard Bayesian framework does not account for the decision maker's errors, such as the conjunction and disjunction fallacies, in updating the belief state upon measuring the uncertain quantities. Moreover, the classical probability theory does not distinguish a mixed state (e.g., being of two minds) from a superposed state (e.g., being of neutral mind). The operator sum representation of completely positive linear maps can address these limitations in a unified manner, and allows for numerical determination of a general belief-state update rule. In particular, examples show that the operator sum approach yields an improvement over the existing quantum model of human inference, and that it can be used to explain the mixing and superposition effects of gossip interactions.