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.
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.
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.
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.
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.
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
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