Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images

A general Thurstonian-type representation (with stochastically interdependent images and probabilistic decisions) for a “same-different” discrimination probability function is a model in which the two stimuli are mapped into two generally interdependent random images and taking on their values in some “perceptual” space; and the realizations of these two random images in a given trial determine the probability with which and in this trial are judged to be different. While stochastically interdependent, and are selectively attributed to (influenced by), respectively, and , which is understood as the possibility of conditioning and on some random variable R that renders them stochastically independent, with their conditional distributions selectively depending on, respectively, and . A general Thurstonian-type representation is considered “well-behaved” if the conditional probability with which and , given a value of the conditioning random variable R, fall within two given subsets of the perceptual space, possess appropriately defined bounded directional derivatives with respect to and . It is shown that no such well-behaved Thurstonian-type representation can account for possessing two basic properties: regular minimality and nonconstant self-similarity. At the same time, an alternative to Thurstonian-type modeling (a model employing “uncertainty blobs” in stimulus spaces instead of random variables in perceptual spaces) is readily available that predicts these two properties “automatically”.     

On the provenance of judgments of conditional probability

In standard treatments of probability, is defined as the ratio of to , provided that . This account of conditional probability suggests a psychological question, namely, whether estimates of arise in the mind via implicit calculation of . We tested this hypothesis (Experiment 1) by presenting brief visual scenes composed of forms, and collecting estimates of relevant probabilities. Direct estimates of conditional probability were not well predicted by . Direct estimates were also closer to the objective probabilities defined by the stimuli, compared to estimates computed from the foregoing ratio. The hypothesis that arises from the ratio fared better (Experiment 2). In a third experiment, the same hypotheses were evaluated in the context of subjective estimates of the chance of future events.     

Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images

A discrimination probability function obtained in the “same-different” paradigm assigns to every ordered pair of stimuli the probability with which they are judged to be different. This function is said to possess the regular minimality property if, for any stimulus pair ,

     

Dissimilarity cumulation theory in arc-connected spaces

This paper continues the development of the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling [Dzhafarov, E.N and Colonius, H. (2007). Dissimilarity Cumulation theory and subjective metrics. Journal of Mathematical Psychology, 51, 290-304]. In arc-connected spaces the notion of a chain length (the sum of the dissimilarities between the chain’s successive elements) can be used to define the notion of a path length, as the limit inferior of the lengths of chains converging to the path in some well-defined sense. The class of converging chains is broader than that of converging inscribed chains. Most of the fundamental results of the metric-based path length theory (additivity, lower semicontinuity, etc.) turn out to hold in the general dissimilarity-based path length theory. This shows that the triangle inequality and symmetry are not essential for these results, provided one goes beyond the traditional scheme of approximating paths by inscribed chains. We introduce the notion of a space with intermediate points which generalizes (and specializes to when the dissimilarity is a metric) the notion of a convex space in the sense of Menger. A space is with intermediate points if for any distinct there is a different point such that (where D is dissimilarity). In such spaces the metric G induced by D is intrinsic: coincides with the infimum of lengths of all arcs connecting to In Universal Fechnerian Scaling D stands for either of the two canonical psychometric increments and (ψ denoting discrimination probability). The choice between the two makes no difference for the notions of arc-connectedness, convergence of chains and paths, intermediate points, and other notions of the Dissimilarity Cumulation theory.     

You’ll see what you mean: Students encode equations based on their knowledge of arithmetic

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth-grade students solved and explained a set of typical addition problems (e.g., ) and mathematical equivalence problems (e.g., or ). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems of the form overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students’ encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank-final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank-final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding.     

Using information criteria to determine the number of regimes in threshold autoregressive models

Threshold autoregressive models can be used to study processes that are characterized by recurrent switches between two or more regimes, where switching is triggered by a manifest threshold variable. In this paper the performance of diverse information criteria for selecting the number of regimes in small to moderate sample sizes (i.e., n=50,100,200) is investigated. In addition it is investigated whether these information criteria can be used to determine whether the residual variances are identical across the regimes. It is concluded that for small sample sizes should be preferred, while for larger sample sizes either BIC or should be considered: The latter is the only information criterion that includes a penalty for the unknown threshold parameters.     

Generalized Pexider equation on a restricted domain

Let X be a normed space and D be a nonempty, open and connected subset of X×X. Inspired by a problem of J. Aczél, we study the functional equation

     

LSD response in Eysenckian trait types identified by polypredictive CFA

The four personality type combinations derived from high and low extraversion () and high and low neuroticism () have been related to response patterns composed of three symptoms (affective disturbances, thinking disturbances, and blackouts) scored as present (+) or absent (−) after a single oral dose of the hallucinogenic drug LSD-25. Hypotheses for expected response patterns for each personality group were derived from a data set obtained by Kohnen and Lienert (1987). Significance of associations was tested by two strategies of polyprediction configural frequency analysis (CFA): multiple uniprediction and biprediction CFA. Both strategies yielded a significant hyperpresentation of all three symptoms present in E+N+ (hysterics), merely thinking disorders in dysthymics (E−N+), merely affective symptoms in E+N− (stable extraverts), and merely blackouts in N−E− (stable introverts). Authors tried to relate these symptoms to Kretschmer's temperament types and could afterwards show by a chessboard modification of prediction CFA, that by applying two combined hypotheses for two personality types each, the significance of the predicted associations could be increased.     

Sequential learning using temporal context

The temporal context model (TCM) has been extensively applied to recall phenomena from episodic memory. Here we use the same formulation of temporal context to construct a sequential learning model called the predictive temporal context model (pTCM) to extract the generating function of a language from sequentially-presented words. In pTCM, temporal context is used to generate a prediction vector at each step of learning and these prediction vectors are in turn used to construct semantic representations of words on the fly. The semantic representation of a word is defined as the superposition of prediction vectors that occur prior to the presentation of the word in the sequence. Here we create a formal framework for pTCM and prove several useful results. We explore the effect of manipulating the parameters of the model on learning a sequence of words generated by a bi-gram generating function. In this simple case, we demonstrate that feeding back the constructed semantic representation into the temporal context during learning improves the performance of the model when trained with a finite training sequence from a language with equivalence classes among some words. We also describe , a variant of the model that is identical to pTCM at steady state. has significant computational advantages over pTCM and can improve the quality of its prediction for some training sequences.     

Functional equations arising in a theory of rank dependence and homogeneous joint receipts

This paper focuses on a class of utility representations of uncertain alternatives with two possible consequences (binary gambles) when they are linked via a distributivity property called segregation to an operation of joint receipt, which may be non-commutative. The assumption that the gambling structure and the joint receipt operation both have homogeneous representations that are order preserving leads to a functional equation that has too many solutions to be useful for characterizing a reasonably specific utility representation. A plausible restriction on the form of the utility of gambles leads to the functional equation