A general theory of measurement context effects, called Hilbert space multidimensional (HSM) theory, is presented. A measurement context refers to a subset of psychological variables that an individual evaluates on a particular occasion. Different contexts are formed by evaluating different but possibly overlapping subsets of variables. Context effects occur when the judgments across contexts cannot be derived from a single joint probability distribution over the complete set of values of the observed variables. HSM theory provides a way to model these context effects by using quantum probability theory, which represents all the variables within a low dimensional vector space. HSM models produce parameter estimates that provide a simple and informative interpretation of the complex collection of judgments across contexts. Comparisons of HSM model fits with Bayesian network model fits are reported for a new large experiment, demonstrating the viability of this new model. We conclude that the theory is broadly applicable to measurement context effects found in the social and behavioral sciences. (PsycINFO Database Record
A general theory of measurement context effects, called Hilbert space multidimensional (HSM) theory, is presented. A measurement context refers to a subset of psychological variables that an individual evaluates on a particular occasion. Different contexts are formed by evaluating different but possibly overlapping subsets of variables. Context effects occur when the judgments across contexts cannot be derived from a single joint probability distribution over the complete set of values of the observed variables. HSM theory provides a way to model these context effects by using quantum probability theory, which represents all the variables within a low dimensional vector space. HSM models produce parameter estimates that provide a simple and informative interpretation of the complex collection of judgments across contexts. Comparisons of HSM model fits with Bayesian network model fits are reported for a new large experiment, demonstrating the viability of this new model. We conclude that the theory is broadly applicable to measurement context effects found in the social and behavioral sciences. (PsycINFO Database Record
This article presents general procedures for constructing, estimating, and testing Hilbert space multi-dimensional (HSM) models, which are based on quantum probability theory. HSM models can be applied to collections of K different contingency tables obtained from a set of p variables that are measured under different contexts. A context is defined by the measurement of a subset of the p variables that are used to form a table. HSM models provide a representation of the collection of K tables in a low dimensional vector space, even when no single joint probability distribution across the p variables exists. HSM models produce parameter estimates that provide a simple and informative interpretation of the complex collection of tables. Comparisons of HSM model fits with Bayes net model fits are reported for a new large experiment, demonstrating the viability of this new model. We conclude that the model is broadly applicable to social and behavioral science data sets.
Data fusion problems arise when a researcher needs to analyse results obtained by measuring empirical variables under different measurement contexts. A context is defined by a subset of variables taken from a complete set of variables under investigation. Multiple contexts can be formed from different subsets, which produce a separate distribution of measurements associated with each context. A context effect occurs when the distributions produced by the different contexts cannot be reproduced by marginalizing over a complete joint distribution formed by all the variables. We propose a Hilbert space multidimensional theory that uses a state vector and measurement operators to account for multiple distributions produced by different contexts. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
Recently Szangolies has argued (in the setting of extended Wigner's-friend scenarios) that quantum theory permits ``Rashomon'' situations: multiple internally coherent accounts of events that cannot be combined into a single, consistent global narrative. This note explains why the Rashomon phenomenon can be understood as a \emph{failure of gluing}: local descriptions over different contexts exist, but they do not admit a single global ``all-perspectives-at-once'' description. This is the same mathematical obstruction that underlies modern sheaf-theoretic treatments of contextuality. I then indicate why the same perspective is useful in parts of the social sciences (quantum-like modelling of cognition, judgment, and decision-making), where ``context effects'' can likewise be interpreted as the absence of a single joint probability space.
Latent variables play a crucial role in psychometric research, yet traditional models often struggle to address context-dependent effects, ambivalent states, and non-commutative measurement processes. This study proposes a quantum-inspired framework for latent variable modeling that employs Hilbert space representations, allowing questionnaire items to be treated as pure or mixed quantum states. By integrating concepts such as superposition, interference, and non-commutative probabilities, the framework captures cognitive and behavioral phenomena that extend beyond the capabilities of classical methods. To illustrate its potential, we introduce quantum-specific metrics—fidelity, overlap, and von Neumann entropy—as complements to correlation-based measures. We also outline a machine-learning pipeline using complex and real-valued neural networks to handle amplitude and phase information. Results highlight the capacity of quantum-inspired models to reveal order effects, ambivalent responses, and multimodal distributions that remain elusive in standard psychometric approaches. This framework broadens the multivariate analysis theoretical and methodological toolkit, offering a dynamic and context-sensitive perspective on latent constructs while inviting further empirical validation in diverse research settings.
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.
In quantum physics, the term ‘contextual’ can be used in more than one way. One usage, here called ‘Bell contextual’ since the idea goes back to Bell, is that if A , B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C , whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the { A , B } context) or with C (the { A , C } context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C . A different definition, here called ‘globally (non)contextual’ refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.