A quantum geometric model 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 (1997) 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 quantum theoretical explanation for probability judgment errors

A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction, disjunction, inverse, and conditional fallacies, as well as unpacking effects and partitioning effects. Quantum probability theory is a general and coherent theory based on a set of (von Neumann) axioms which relax some of the constraints underlying classic (Kolmogorov) probability theory. The quantum model is compared and contrasted with other competing explanations for these judgment errors including the representativeness heuristic, the averaging model, and a memory retrieval model for probability judgments. The quantum model also provides ways to extend Bayesian, fuzzy set, and fuzzy trace theories. We conclude that quantum information processing principles provide a viable and promising new way to understand human judgment and reasoning

Structured representations in a quantum probability model of similarity

Recently, Busemeyer et al. (2011) presented a model for how the conjunction fallacy (Tversky & Kahneman, 1983) emerges, based on the principles of quantum probability (QP) theory. Pothos et al. (2013) extended this model to account for the main similarity findings of Tversky (1977), which have served as a golden standard for testing novel theories of similarity. However, Tversky’s (1977) empirical findings did not address the now established insight that, in comparing two objects, overlap in matching parts of the objects tends to have a greater impact on their similarity, than overlap in non-matching parts. We show how the QP similarity model can be directly extended to accommodate structure in similarity comparisons. Smolensky’s et al.’s (2014) proposal for modeling structure in linguistic representations, with tensor products, can be adapted ‘as is’ with the QP similarity model. The formal properties of the extended QP similarity model are analyzed, some indicative fits are presented, and, finally, a novel prediction is developed

The conjunction fallacy, confirmation, and quantum theory: comment on Tentori, Crupi, & Russo

The conjunction fallacy refers to situations when a person judges a conjunction to be more likely than one of the individual conjuncts, which is a violation of a key property of classical probability theory. Recently, quantum probability theory has been proposed as a coherent account of these and many other findings on probability judgment “errors” that violate classical probability rules, including the conjunction fallacy. Tentori, Crupi, and Russo (2013) present an alternative account of the conjunction fallacy based on the concept of inductive confirmation. They present new empirical findings consistent with their account, and they also claim that these results are inconsistent with the quantum probability theory account. This comment proves that our quantum probability model for the conjunction fallacy is completely consistent with the main empirical results from Tentori et al. (2013). Furthermore, we discuss experimental tests that can distinguish the two alternative accounts

A quantum probability account of individual differences in causal reasoning

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

Quantum probability updating from zero priors (by-passing Cromwell’s rule)

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

A quantum theory account of order effects and conjunction fallacies in political judgments

Are our everyday judgments about the world around us normative? Decades of research in the judgment and decision-making literature suggest the answer is no. If people's judgments do not follow normative rules, then what rules if any do they follow? Quantum probability theory is a promising new approach to modeling human behavior that is at odds with normative, classical rules. One key advantage of using quantum theory is that it explains multiple types of judgment errors using the same basic machinery, unifying what have previously been thought of as disparate phenomena. In this article, we test predictions from quantum theory related to the co-occurrence of two classic judgment phenomena, order effects and conjunction fallacies, using judgments about real-world events (related to the U.S. presidential primaries). We also show that our data obeys two a priori and parameter free constraints derived from quantum theory. Further, we examine two factors that moderate the effects, cognitive thinking style (as measured by the Cognitive Reflection Test) and political ideology

Modeling the Frozen-In Anticyclone in the 2005 Arctic Summer Stratosphere

Immediately following the breakup of the 2005 Arctic spring stratospheric vortex, a tropical air mass, characterized by low potential vorticity (PV) and high nitrous oxide (N2O), was advected poleward and became trapped in the easterly summer polar vortex. This feature, known as a "Frozen-In Anticyclone (FrIAC)", was observed in Earth Observing System (EOS) Aura Microwave Limb Sounder (MLS) data to span the potential temperature range from approximately 580 to 1100 K (approximately 25 to 40 km altitude) and to persist from late March to late August 2005. This study compares MLS N2O observations with simulations from the Global Modeling Initiative (GMI) chemistry and transport model, the GEOS-5/MERRA Replay model, and the VanLeer Icosahedral Triangular Advection isentropic transport model to elucidate the processes involved in the lifecycle of the FrIAC which is here divided into three distinct phases. During the "spin-up phase" (March to early April), strong poleward flow resulted in a tight isolated anticyclonic vortex at approximately 70-90 deg N, marked with elevated N2O. GMI, Replay, and VITA all reliably simulted the spin-up of the FrIAC, although the GMI and Replay peak N2O values were too low. The FrIAC became trapped in the developing summer easterly flow and circulated around the polar region during the "anticyclonic phase" (early April to the end of May). During this phase, the FrIAC crossed directly over the pole between the 7th and 14th of April. The VITA and Replay simulations transported the N2O anomaly intact during this crossing, in agreement with MLS, but unrealistic dispersion of the anomaly occurred in the GMI simulation due to excessive numerical mixing of the polar cap. The vortex associated with the FrIAC was apparently resistant to the weak vertical hear during the anticyclonic phase, and it thereby protected the embedded N20 anomaly from stretching. The vortex decayed in late May due to diabatic processes, leaving the N2O anomaly exposed to horizontal and vertical wind shears during the "shearing phase" (June to August). The observed lifetime of the FrIAC during this phase is consistent with time-scales calculated from the ambient horizontal and vertical wind shear. Replay maintained the horizontal structure of the N2O anomaly similar to NILS well into August. The VITA simulation also captured the horizontal structure of the FrIAC during this phase, but VITA eventually developed fine-scale N2O structure not observed in MLS data

KELT-3b: A Hot Jupiter Transiting A V=9.8 Late-F Star

We report the discovery of KELT-3b, a moderately inflated transiting hot Jupiter with a mass of 1.477(-0.067)(+0.066) M-J, radius of 1.345 +/- 0.072 R-J, and an orbital period of 2.7033904 +/- 0.000010 days. The host star, KELT-3, is a V = 9.8 late F star with M-* = 1.278(-0.061)(+0.063) M-circle dot, R-* = 1.472(-0.067)(+0.065) R-circle dot, T-eff = 6306(-49)(+50) K, log(g) = 4.209(-0.031)(+0.033), and [Fe/H] = 0.044(-0.082)(+0.080), and has a likely proper motion companion. KELT-3b is the third transiting exoplanet discovered by the KELT survey, and is orbiting one of the 20 brightest known transiting planet host stars, making it a promising candidate for detailed characterization studies. Although we infer that KELT-3 is significantly evolved, a preliminary analysis of the stellar and orbital evolution of the system suggests that the planet has likely always received a level of incident flux above the empirically identified threshold for radius inflation suggested by Demory & Seager

KELT-20b: A Giant Planet With A Period Of P ~ 3.5 Days Transiting The V ~ 7.6 Early A Star HD 185603

We report the discovery of KELT-20b, a hot Jupiter transiting a early A star, HD 185603, with an orbital period of days. Archival and follow-up photometry, Gaia parallax, radial velocities, Doppler tomography, and AO imaging were used to confirm the planetary nature of KELT-20b and characterize the system. From global modeling we infer that KELT-20 is a rapidly rotating ( ) A2V star with an effective temperature of K, mass of , radius of , surface gravity of , and age of . The planetary companion has a radius of , a semimajor axis of au, and a linear ephemeris of . We place a upper limit of on the mass of the planet. Doppler tomographic measurements indicate that the planetary orbit normal is well aligned with the projected spin axis of the star ( ). The inclination of the star is constrained to , implying a three-dimensional spin–orbit alignment of . KELT-20b receives an insolation flux of , implying an equilibrium temperature of of ∼2250 K, assuming zero albedo and complete heat redistribution. Due to the high stellar , KELT-20b also receives an ultraviolet (wavelength nm) insolation flux of , possibly indicating significant atmospheric ablation. Together with WASP-33, Kepler-13 A, HAT-P-57, KELT-17, and KELT-9, KELT-20 is the sixth A star host of a transiting giant planet, and the third-brightest host (in V ) of a transiting planet