Since Tversky argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been particularly challenging for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky, asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize, especially for perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments, we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. proposed a quantum geometric model of similarity to account for Tversky's findings. In the present work, we extended this model to a more general framework that can be fit to similarity judgments. We fitted several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the best-fit quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric framework of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.
Since Tversky argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been particularly challenging for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky, asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize, especially for perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments, we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. proposed a quantum geometric model of similarity to account for Tversky's findings. In the present work, we extended this model to a more general framework that can be fit to similarity judgments. We fitted several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the best-fit quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric framework of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.
Since Tversky (1977) argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been offered as particularly difficult challenges for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky (1977), asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize to different kinds of stimuli, especially perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. (2013) proposed a quantum geometric model of similarity to account for Tversky’s (1977) findings. In the present work, we extended this model to a more general framework that can be fit to similarity judgments. We fit several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the best-fit quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric framework of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.
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.
A model of similarity is presented which is based on Quantum Probability (QP) theory. The model is applied to the case of violations of symmetry in similarity judgments, as demonstrated by Tversky (1977). The QP similarity model can predict such violations, on the basis of the same underlying intuitions as Tversky (1977). Moreover, we discuss how the model can be extended to account for violations of the triangle inequality and also the empirical findings in relation to Tversky’s diagnosticity principle.
In alignment with the distributional hypothesis of language, the work “Quantum Projections on Conceptual Subspaces” (Martínez-Mingo A, Jorge-Botana G, Martinez-Huertas JÁ, et al. Quantum projections on conceptual subspaces. Cogn Syst Res 2023; 82: 101154) proposed a methodology for generating conceptual subspaces from textual information based on previous work (Martinez-Mingo A, Jorge-Botana G and Olmos R. Quantum approach for similarity evaluation in LSA vector space models. 2020). These subspaces enable the utilization of the quantum model of similarity put forth by Pothos and Busemeyer (Pothos E, Busemeyer J. A quantum probability explanation for violations of symmetry in similarity judgments. In Proceedings of the annual meeting of the cognitive science society, 2011, Vol. 33, No. 33), allowing for the empirical examination of the violations of assumptions concerning symmetry and triangular inequality (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327–352; Yearsley JM, Barque-Duran A, Scerrati E, et al. The triangle inequality constraint in similarity judgments. Prog Biophys Mol Biol 2017; 130: 26–32), as well as the diagnosticity effect (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327–352; Yearsley JM, Pothos EM, Barque-Duran A, et al. Context effects in similarity judgments. J Exp Psychol Gen 2022; 151: 711–717), within a data-driven environment. These psychological biases, deeply studied by authors such as Tversky and Kahneman, inform us about the limitations of modeling psychological similarity measures using tools from classical geometry. This commentary aims to offer methodological clarifications, discuss theoretical and practical implications, and speculate on future directions in this field of research. Concretely, it aims to propose the use of different contours (conceptual or contextual) to generate the subspaces, which lead to subspaces of terms or contexts. Once these contours are defined, a differentiation is proposed between Aggregated Terms Subspaces (ATSs), Aggregated Contexts Subspaces (ACSs), and Aggregated Features Subspaces (AFSs) depending on whether we define the subspaces by grouping the terms or contexts within the contour, or from the latent dimensions of the semantic space obtained in the contour window. Finally, new data is provided on the violation of the triangular inequality assumption through the application of the quantum similarity model to ATSs.
This paper presents a novel theoretical framework centered around the concept of "Quantum Time Scale Geometry" – a methodology for treating time as a fundamental geometric element and applying this perspective to probabilistic modeling. We argue that time, traditionally viewed as a sequential and dynamic process, can be fundamentally represented as a geometric structure, offering a powerful tool for precisely predicting complex system behavior. The core mechanism involves manipulating time scales through geometric transformations, enabling more accurate probability predictions than conventional approaches. We explore the implications of this framework for both probability theory and statistical modeling, highlighting its potential to address longstanding challenges in understanding and predicting the behavior of complex systems. This work offers a fresh perspective on time, potentially revolutionizing approaches to modeling phenomena involving temporal dynamics and uncertainty.
We present progress towards a novel theoretical approach for\nunderstanding Tversky’s famous ‘diagnosticity’ effect in similarity\njudgments, and an initial empirical validation. Our approach\nuses a model for similarity judgments based on quantum\nprobability theory. The model predicts a diagnosticity effect\nunder certain conditions only. Our model also predicts that\nchanges to the set of stimuli to be compared can cause the diagnosticity\neffect to break down or reverse. In one experiment,\nwe test and confirm one of our key predictions