“Pseudodiagnostic" patterns of information search were first classified by Doherty et al. (1979), predicated on their assertion that only the selection of data that allow for the calculation of Bayseian probability ratios may be considered rational. However, with the exception of Crupi et al. (2009), who have argued for an epistemological explanation loosely based on the de Finetti theorem of exchangeable probability assignments (see, eg., Heath and Sudderth, 1972), the Doherty et al. interpretation of information search patterns has gone unchallenged. This thesis seeks to answer three questions: can people make reasonable inferences from probabilistic information; is there an identifiable common approach to information selection; and, if not Bayes' theorem, then what mathematics might underlie human decision-making? A series of experiments demonstrate that people appear to select data not only for their ordinal values, but also to establish the relationships between them. However, since such a holistic approach to decision-making lies beyond the scope of the naïve Bayes' classifier, an alternative expression for the calculation of likelihood ratios is presented. By rejecting the Kolmogorov axioms of classical statistics in favour of the von Neumann mathematical axioms of quantum mechanics, it is shown that not only may probabilistic information be reconceptualised as isomorphic representations of quantised and entangled statistical systems, but that it is only this approach which allows for the assumption free calculation of likelihood ratios. Further, the mathematical derivation demonstrates that Bayes' theorem is a special case of this more general quantum expression, applicable only where the conditional independence of data is guaranteed. Mathematical modelling, combined with the results of another experiment, is used to investigate the plausibility of this expression as an explanation for both information search patterns, and people's estimation of probability. The nature of this “relational information theory" is both discussed and situated within the wider psychological field of mental models.
“Pseudodiagnostic" patterns of information search were first classified by Doherty et al. (1979), predicated on their assertion that only the selection of data that allow for the calculation of Bayseian probability ratios may be considered rational. However, with the exception of Crupi et al. (2009), who have argued for an epistemological explanation loosely based on the de Finetti theorem of exchangeable probability assignments (see, eg., Heath and Sudderth, 1972), the Doherty et al. interpretation of information search patterns has gone unchallenged. This thesis seeks to answer three questions: can people make reasonable inferences from probabilistic information; is there an identifiable common approach to information selection; and, if not Bayes' theorem, then what mathematics might underlie human decision-making? A series of experiments demonstrate that people appear to select data not only for their ordinal values, but also to establish the relationships between them. However, since such a holistic approach to decision-making lies beyond the scope of the naïve Bayes' classifier, an alternative expression for the calculation of likelihood ratios is presented. By rejecting the Kolmogorov axioms of classical statistics in favour of the von Neumann mathematical axioms of quantum mechanics, it is shown that not only may probabilistic information be reconceptualised as isomorphic representations of quantised and entangled statistical systems, but that it is only this approach which allows for the assumption free calculation of likelihood ratios. Further, the mathematical derivation demonstrates that Bayes' theorem is a special case of this more general quantum expression, applicable only where the conditional independence of data is guaranteed. Mathematical modelling, combined with the results of another experiment, is used to investigate the plausibility of this expression as an explanation for both information search patterns, and people's estimation of probability. The nature of this “relational information theory" is both discussed and situated within the wider psychological field of mental models.
The ability to calculate precise likelihood ratios is fundamental to science, from Quantum Information Theory through to Quantum State Estimation. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes' theorem either defaults to the marginal probability driven "naive Bayes' classifier", or requires the use of compensatory expectation-maximization techniques. This article takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement, and demonstrates that Bayes' theorem is a special case of a more general quantum mechanical expression.
The ability to calculate precise likelihood ratios is fundamental to science, from Quantum Information Theory through to Quantum State Estimation. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes’ theorem either defaults to the marginal probability driven “naive Bayes’ classifier”, or requires the use of compensatory expectation-maximization techniques. This paper takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement, and demonstrates that Bayes’ theorem is a special case of a more general quantum mechanical expression.
The present thesis (mostly) concerns the application of alternative mathematical methods to understand patterns in human cognition and to model them. The different chapters presented in this thesis show research that concerns the application of quantum probability (QP) theory in the modeling of human decision-making. Quantum probability (QP) theory is a theory for how to assign probabilities to events. QP theory can be thought of as the probability rules from quantum mechanics, without any of the physics. This work is not about the application of quantum physics to brain physiology. Rather, we are interested in QP theory as a mathematical framework for cognitive modelling. This theory is potentially relevant in any behavioural situation that involves uncertainty. QP theory is analogous to classical probability theory, though QP theory and classical probability (CP) theory are founded from different sets of axioms (the Kolmogorov and Dirac/von Neumann axioms respectively) and so are subject to alternative constraints. In this thesis we show that especially over the last decade, there has been a growing interest in decision-making and cognitive models using a quantum probabilistic (QP) framework. We see how this development encompasses publications in major journals (see Pothos and Busemeyer, 2013; Wang et al., 2014; and Yearsley and Pothos, 2014; among others), special issues, and dedicated workshops, as well as several comprehensive books (Busemeyer and Bruza, 2012; Khrennikov, 2010; and Haven and Khrennikov, 2010). However, uncertainty itself is neither ethical nor unethical – yet it is inherent to most situations in which, for instance, moral judgments and decisions have to be made. For a descriptive understanding of judgment and decisions in moral situations, it is an important lesson to acknowledge both the cognitive side (bounded rationality) and the environment (ecological rationality) – and thus the uncertainty of the world and how the mind deals with it. This thesis also shows significant interest in moral and social psychology. Specifically, we consider present technologies that suggest a need for evaluating alternative contexts for ethical decision-making. How the research on human-machine interaction feeds back into humans’ understanding of themselves as moral agents? This key question ultimately relates to the nature of ethical theory itself. Overall, this dissertation presents and addresses not only standard aspects of decision- making processes, such as similarity judgments (Chapters 1 to 4) or the constructive role of articulating impressions (Chapter 5), but also standard aspects of social psychology, such as moral judgments (Chapters 6 and 7) and game theory (Chapter 8). As stated in the Declarations section, the present thesis is a combination of a standard and a publication- based dissertation.
There is considerable variety in human inference (e.g., a doctor inferring the presence of a disease, a juror inferring the guilt of a defendant, or someone inferring future weight loss based on diet and exercise). As such, people display a wide range of behaviors when making inference judgments. Sometimes, people's judgments appear Bayesian (i.e., normative), but in other cases, judgments deviate from the normative prescription of classical probability theory. How can we combine both Bayesian and non-Bayesian influences in a principled way? We propose a unified explanation of human inference using quantum probability theory. In our approach, we postulate a hierarchy of mental representations, from 'fully' quantum to 'fully' classical, which could be adopted in different situations. In our hierarchy of models, moving from the lowest level to the highest involves changing assumptions about compatibility (i.e., how joint events are represented). Using results from 3 experiments, we show that our modeling approach explains 5 key phenomena in human inference including order effects, reciprocity (i.e., the inverse fallacy), memorylessness, violations of the Markov condition, and antidiscounting. As far as we are aware, no existing theory or model can explain all 5 phenomena. We also explore transitions in our hierarchy, examining how representations change from more quantum to more classical. We show that classical representations provide a better account of data as individuals gain familiarity with a task. We also show that representations vary between individuals, in a way that relates to a simple measure of cognitive style, the Cognitive Reflection Test. (PsycINFO Database Record
Experiments in cognitive science and decision theory show that the ways in which people combine concepts and make decisions cannot be described by classical logic and probability theory. This has serious implications for applied disciplines such as information retrieval, artificial intelligence and robotics. Inspired by a mathematical formalism that generalizes quantum mechanics the authors have constructed a contextual framework for both concept representation and decision making, together with quantum models that are in strong alignment with experimental data. The results can be interpreted by assuming the existence in human thought of a double-layered structure, a 'classical logical thought' and a 'quantum conceptual thought', the latter being responsible of the above paradoxes and nonclassical effects. The presence of a quantum structure in cognition is relevant, for it shows that quantum mechanics provides not only a useful modeling tool for experimental data but also supplies a structural model for human and artificial thought processes. This approach has strong connections with theories formalizing meaning, such as semantic analysis, and has also a deep impact on computer science, information retrieval and artificial intelligence. More specifically, the links with information retrieval are discussed in this paper.
After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.