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

A Comparison of Interpolation Methods for Sparse Data: Application to Wind and Concentration Fields

In order to produce gridded fields of pollutant concentration data and surface wind data for use in an air quality model, a number of techniques for interpolating sparse data values are compared. The techniques are compared using three data sets. One is an idealized concentration distribution to which the exact solution is known, the second is a potential flow field, while the third consists of surface ozone concentrations measured in the Los Angeles Basin on a particular day. The results of the study indicate that fitting a second-degree polynomial to each subregion (triangle) in the plane with each data point weighted according to its distance from the subregion provides a good compromise between accuracy and computational cost

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

Geography and Giving: The Culture of Philanthropy in New England and the Nation

Looks at aggregate household wealth and income at the national level and for Massachusetts as a state, and analyzes levels of charitable giving in relation to household income

A model of adaptive decision making from representation of information environment by quantum fields

We present the mathematical model of decision making (DM) of agents acting in a complex and uncertain environment (combining huge variety of economical, financial, behavioral, and geo-political factors). To describe interaction of agents with it, we apply the formalism of quantum field theory (QTF). Quantum fields are of the purely informational nature. The QFT-model can be treated as a far relative of the expected utility theory, where the role of utility is played by adaptivity to an environment (bath). However, this sort of utility-adaptivity cannot be represented simply as a numerical function. The operator representation in Hilbert space is used and adaptivity is described as in quantum dynamics. We are especially interested in stabilization of solutions for sufficiently large time. The outputs of this stabilization process, probabilities for possible choices, are treated in the framework of classical DM. To connect classical and quantum DM, we appeal to Quantum Bayesianism (QBism). We demonstrate the quantum-like interference effect in DM which is exhibited as a violation of the formula of total probability and hence the classical Bayesian inference scheme.Comment: in press in Philosophical Transactions

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

Measured Spectra of the Hygroscopic Fraction of Atmospheric Aerosol Particles

The relation between dry diameter (X0) and critical supersaturation (Sc) for atmospheric submicron aerosol particles is investigated using a long term air sampling program at Rolla, Missouri. The particles are passed through an electrostatic aerosol size classifier, and then through an isothermal haze chamber. Results are reported in terms of an apparent volume fraction of soluble material, εv defined such that for particles composed only of ammonium sulfate and water insoluble compounds, εv is the actual volume fraction of soluble material. The probability distribution of εv is found to be approximately Gaussian in the εv range 0.2 to 1.3. The mean εv is 0.5, for electrostatic aerosol classifier settings of 0.2, 0.3, and 0.4 μm diameter

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