The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Jerome R Busemeyer - One of the best experts on this subject based on the ideXlab platform.
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introduction to hilbert space multi dimensional modeling
2019Co-Authors: Jerome R Busemeyer, Zheng Joyce WangAbstract:This chapter provides a brief introduction to procedures for estimating Hilbert space multi-dimensional (HSM) models from data. These models, which are built from Quantum Probability theory, are used to provide a simple and coherent account of a collection of contingency tables. The collection of tables are obtained by measurement of different overlapping subsets of variables. HSM models provide a representation of the collection of the tables in a low dimensional vector space, even when no single joint Probability distribution across the observed variables can reproduce the tables. The parameter estimates from HSM models provide simple and informative interpretation of the initial tendencies and the inter-relations among the variables.
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Quantum Probability updating from zero priors by passing cromwell s rule
Journal of Mathematical Psychology, 2017Co-Authors: Irina Basieva, Andrei Khrennikov, Emmanuel M Pothos, Jennifer S Trueblood, Jerome R BusemeyerAbstract:Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical Probability theory that if one assigns a prior Probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum Probability theory, the rules for how to assign probabilities from the mathematical formalism of Quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of Quantum Probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on Quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.
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bayesian model comparison favors Quantum over standard decision theory account of dynamic inconsistency
Decision, 2015Co-Authors: Jerome R Busemeyer, Zheng Wang, Richard M ShiffrinAbstract:Many paradoxical findings in decision-making that have resisted explanations by standard decision theories have accumulated over the past 50 years. Recent advances based on Quantum Probability theory have successfully accounted for many of these puzzling findings. Critics, however, claim that Quantum Probability theory is less constrained than standard Probability theory, and hence Quantum models only fit better because they are more complex than standard decision models. In this article, for the first time, a Bayesian method was used to quantitatively compare the 2 types of decision models, which is a method that evaluates models with respect to accuracy, parsimony, and robustness. A large experiment was used to compare the best-known models of each type, matching in their numbers of parameters, but possibly differing in the complexity of their functional forms. Surprisingly, the Bayesian model comparison overwhelmingly favored the Quantum model, indicating that its success is due to its robust ability to make accurate predictions rather than accidental fits afforded by increased complexity.
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a Quantum geometric model of similarity
Psychological Review, 2013Co-Authors: Emmanuel M Pothos, Jerome R Busemeyer, Jennifer S TruebloodAbstract:No other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tversky's main findings. Similarity is modeled as Quantum Probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of Quantum Probability. We so demonstrate the promise of the Quantum approach for similarity and discuss the implications for representation theory in general.
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can Quantum Probability provide a new direction for cognitive modeling
Behavioral and Brain Sciences, 2013Co-Authors: Emmanuel M Pothos, Jerome R BusemeyerAbstract: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.
Jennifer S Trueblood - One of the best experts on this subject based on the ideXlab platform.
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a Quantum Probability account of individual differences in causal reasoning
Journal of Mathematical Psychology, 2018Co-Authors: Percy K Mistry, Emmanuel M Pothos, Joachim Vandekerckhove, Jennifer S TruebloodAbstract:We use Quantum Probability (QP) theory to investigate individual differences in causal reasoning. By analyzing data sets from Rehder (2014) on comparative judgments, and from Rehder & Waldmann (2016) on absolute judgments, we show that a QP model can both account for individual differences in causal judgments, and why these judgments sometimes violate the properties of causal Bayes nets. We implement this and previously proposed models of causal reasoning (including classical Probability models) within the same hierarchical Bayesian inferential framework to provide a detailed comparison between these models, including computing Bayes factors. Analysis of the inferred parameters of the QP model illustrates how these can be interpreted in terms of putative cognitive mechanisms of causal reasoning. Additionally, we implement a latent classification mechanism that identifies subcategories of reasoners based on properties of the inferred cognitive process, rather than post hoc clustering. The QP model also provides a parsimonious explanation for aggregate behavior, which alternatively can only be explained by a mixture of multiple existing models. Investigating individual differences through the lens of a QP model reveals simple but strong alternatives to existing explanations for the dichotomies often observed in how people make causal inferences. These alternative explanations arise from the cognitive interpretation of the parameters and structure of the Quantum Probability model.
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Quantum Probability updating from zero priors by passing cromwell s rule
Journal of Mathematical Psychology, 2017Co-Authors: Irina Basieva, Andrei Khrennikov, Emmanuel M Pothos, Jennifer S Trueblood, Jerome R BusemeyerAbstract:Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical Probability theory that if one assigns a prior Probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum Probability theory, the rules for how to assign probabilities from the mathematical formalism of Quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of Quantum Probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on Quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.
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a Quantum geometric model of similarity
Psychological Review, 2013Co-Authors: Emmanuel M Pothos, Jerome R Busemeyer, Jennifer S TruebloodAbstract:No other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tversky's main findings. Similarity is modeled as Quantum Probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of Quantum Probability. We so demonstrate the promise of the Quantum approach for similarity and discuss the implications for representation theory in general.
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a Quantum Probability model of causal reasoning
Frontiers in Psychology, 2012Co-Authors: Jennifer S Trueblood, Jerome R BusemeyerAbstract: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.
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a Quantum Probability account of order effects in inference
Cognitive Science, 2011Co-Authors: Jennifer S Trueblood, Jerome R BusemeyerAbstract:Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a Quantum inference model for order effects based on the axiomatic principles of Quantum Probability theory. The Quantum inference model explains order effects by transforming a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the Quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the Quantum inference model, a new jury decision-making experiment is developed. Using the results of this experiment, we compare the Quantum inference model with two versions of the belief-adjustment model, the adding model and the averaging model. We show that both the Quantum model and the adding model provide good fits to the data. To distinguish the Quantum model from the adding model, we develop a new experiment involving extreme evidence. The results from this new experiment suggest that the adding model faces limitations when accounting for tasks involving extreme evidence, whereas the Quantum inference model does not. Ultimately, we argue that the Quantum model provides a more coherent account for order effects that was not possible before.
Massimiliano Sassoli De Bianchi - One of the best experts on this subject based on the ideXlab platform.
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the unreasonable success of Quantum Probability ii Quantum measurements as universal measurements
Journal of Mathematical Psychology, 2015Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:Abstract In the first part of this two-part article (Aerts & Sassoli de Bianchi, 2014), we have introduced and analyzed a multidimensional model, called the general tension-reduction (GTR) model, able to describe general Quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement . In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since Quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied–mutatis mutandis–to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since Quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the Quantum statistics in the description of a large class of phenomena.
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the unreasonable success of Quantum Probability i Quantum measurements as uniform fluctuations
Journal of Mathematical Psychology, 2015Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:Abstract We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science–from cognitive science to physics–and use it to explain the emergence of Quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although the model we propose, which we call the uniform tension-reduction (UTR) model, is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to “release the tension” and produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the UTR-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the general tension-reduction (GTR) model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement , describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing Quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of Quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of Quantum Probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of Quantum cognition, a research field in cognitive science employing the Quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part (Aerts and Sassoli de Bianchi, 2014a), the proof of the equivalence between universal measurements and uniform measurements, and its significance for Quantum theory as a first order approximation, is given and further analyzed.
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the unreasonable success of Quantum Probability i Quantum measurements as uniform fluctuations
arXiv: Quantum Physics, 2014Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:We introduce a 'uniform tension-reduction' (UTR) model, which allows to represent the probabilities associated with an arbitrary measurement situation and use it to explain the emergence of Quantum probabilities (the Born rule) as 'uniform' fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states, in a way that the measurement probabilities can be derived as the 'Lebesgue measure' of suitably defined convex subregions of the simplexes. We consider a very simple and evocative physical realization of the abstract model, using a material point particle which is acted upon by elastic membranes, which by breaking and collapsing produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of an arbitrary measurement process. We also show that the UTR-model can be further generalized into a 'general tension-reduction' (GTR) model, describing conditions of lack of knowledge generated by 'non-uniform' fluctuations. In this ampler framework, particularly suitable to describe experiments in cognitive science, we define and motivate a notion of 'universal measurement', describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing Quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of Quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of Quantum Probability in the description of different domains of reality. This is the first part of a two-part article.
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the unreasonable success of Quantum Probability ii Quantum measurements as universal measurements
arXiv: Quantum Physics, 2014Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the 'general tension-reduction' (GTR) model, able to describe general Quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a 'universal measurement'. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by 'uniform' fluctuations on the measurement situation. Since Quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied -- mutatis mutandis -- to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally 'robust' descriptions of indeterministic reproducible experiments, and since Quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the Quantum statistics in the description of a large class of phenomena.
Emmanuel M Pothos - One of the best experts on this subject based on the ideXlab platform.
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a Quantum Probability account of individual differences in causal reasoning
Journal of Mathematical Psychology, 2018Co-Authors: Percy K Mistry, Emmanuel M Pothos, Joachim Vandekerckhove, Jennifer S TruebloodAbstract:We use Quantum Probability (QP) theory to investigate individual differences in causal reasoning. By analyzing data sets from Rehder (2014) on comparative judgments, and from Rehder & Waldmann (2016) on absolute judgments, we show that a QP model can both account for individual differences in causal judgments, and why these judgments sometimes violate the properties of causal Bayes nets. We implement this and previously proposed models of causal reasoning (including classical Probability models) within the same hierarchical Bayesian inferential framework to provide a detailed comparison between these models, including computing Bayes factors. Analysis of the inferred parameters of the QP model illustrates how these can be interpreted in terms of putative cognitive mechanisms of causal reasoning. Additionally, we implement a latent classification mechanism that identifies subcategories of reasoners based on properties of the inferred cognitive process, rather than post hoc clustering. The QP model also provides a parsimonious explanation for aggregate behavior, which alternatively can only be explained by a mixture of multiple existing models. Investigating individual differences through the lens of a QP model reveals simple but strong alternatives to existing explanations for the dichotomies often observed in how people make causal inferences. These alternative explanations arise from the cognitive interpretation of the parameters and structure of the Quantum Probability model.
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Quantum Probability updating from zero priors by passing cromwell s rule
Journal of Mathematical Psychology, 2017Co-Authors: Irina Basieva, Andrei Khrennikov, Emmanuel M Pothos, Jennifer S Trueblood, Jerome R BusemeyerAbstract:Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical Probability theory that if one assigns a prior Probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum Probability theory, the rules for how to assign probabilities from the mathematical formalism of Quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of Quantum Probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on Quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.
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a Quantum Probability perspective on borderline vagueness
Topics in Cognitive Science, 2013Co-Authors: Reinhard Blutner, Emmanuel M Pothos, Peter BruzaAbstract:The term “vagueness” describes a property of natural concepts, which normally have fuzzy boundaries, admit borderline cases, and are susceptible to Zeno’s sorites paradox. We will discuss the psychology of vagueness, especially experiments investigating the judgment of borderline cases and contradictions. In the theoretical part, we will propose a probabilistic model that describes the quantitative characteristics of the experimental finding and extends Alxatib’s and Pelletier’s (2011) theoretical analysis. The model is based on a Hopfield network for predicting truth values. Powerful as this classical perspective is, we show that it falls short of providing an adequate coverage of the relevant empirical results. In the final part, we will argue that a substantial modification of the analysis put forward by Alxatib and Pelletier and its probabilistic pendant is needed. The proposed modification replaces the standard notion of probabilities by Quantum probabilities. The crucial phenomenon of borderline contradictions can be explained then as a Quantum interference phenomenon.
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a Quantum geometric model of similarity
Psychological Review, 2013Co-Authors: Emmanuel M Pothos, Jerome R Busemeyer, Jennifer S TruebloodAbstract:No other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tversky's main findings. Similarity is modeled as Quantum Probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of Quantum Probability. We so demonstrate the promise of the Quantum approach for similarity and discuss the implications for representation theory in general.
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can Quantum Probability provide a new direction for cognitive modeling
Behavioral and Brain Sciences, 2013Co-Authors: Emmanuel M Pothos, Jerome R BusemeyerAbstract: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.
Diederik Aerts - One of the best experts on this subject based on the ideXlab platform.
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the unreasonable success of Quantum Probability ii Quantum measurements as universal measurements
Journal of Mathematical Psychology, 2015Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:Abstract In the first part of this two-part article (Aerts & Sassoli de Bianchi, 2014), we have introduced and analyzed a multidimensional model, called the general tension-reduction (GTR) model, able to describe general Quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement . In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since Quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied–mutatis mutandis–to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since Quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the Quantum statistics in the description of a large class of phenomena.
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the unreasonable success of Quantum Probability i Quantum measurements as uniform fluctuations
Journal of Mathematical Psychology, 2015Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:Abstract We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science–from cognitive science to physics–and use it to explain the emergence of Quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although the model we propose, which we call the uniform tension-reduction (UTR) model, is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to “release the tension” and produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the UTR-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the general tension-reduction (GTR) model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement , describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing Quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of Quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of Quantum Probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of Quantum cognition, a research field in cognitive science employing the Quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part (Aerts and Sassoli de Bianchi, 2014a), the proof of the equivalence between universal measurements and uniform measurements, and its significance for Quantum theory as a first order approximation, is given and further analyzed.
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the unreasonable success of Quantum Probability i Quantum measurements as uniform fluctuations
arXiv: Quantum Physics, 2014Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:We introduce a 'uniform tension-reduction' (UTR) model, which allows to represent the probabilities associated with an arbitrary measurement situation and use it to explain the emergence of Quantum probabilities (the Born rule) as 'uniform' fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states, in a way that the measurement probabilities can be derived as the 'Lebesgue measure' of suitably defined convex subregions of the simplexes. We consider a very simple and evocative physical realization of the abstract model, using a material point particle which is acted upon by elastic membranes, which by breaking and collapsing produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of an arbitrary measurement process. We also show that the UTR-model can be further generalized into a 'general tension-reduction' (GTR) model, describing conditions of lack of knowledge generated by 'non-uniform' fluctuations. In this ampler framework, particularly suitable to describe experiments in cognitive science, we define and motivate a notion of 'universal measurement', describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing Quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of Quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of Quantum Probability in the description of different domains of reality. This is the first part of a two-part article.
-
the unreasonable success of Quantum Probability ii Quantum measurements as universal measurements
arXiv: Quantum Physics, 2014Co-Authors: Diederik Aerts, Massimiliano Sassoli De BianchiAbstract:In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the 'general tension-reduction' (GTR) model, able to describe general Quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a 'universal measurement'. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by 'uniform' fluctuations on the measurement situation. Since Quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied -- mutatis mutandis -- to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally 'robust' descriptions of indeterministic reproducible experiments, and since Quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the Quantum statistics in the description of a large class of phenomena.