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”.     

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 ,

     

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.     

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.     

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

     

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.     

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

     

Dissimilarity cumulation theory and subjective metrics

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a≠b?Dab>0; (2) Daa=0; (3) If and , then ; and (4) for any sequence {a_nX_nb_n}_n∈N, where X_n is a chain of stimuli, Da_nX_nb_n→0?Da_nb_n→0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb+DbYa across all possible chains X and Y inserted between a and b.     

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.     

Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective

This study employed conservation of resources (CORs) theory to propose and test relationships between resources possessed by employees, their coping strategies, and emotional exhaustion. The participants consisted of 600 full-time government employees. An OLS regression showed that, in general and consistent with COR theory, resource levels were positively associated with the use of active coping strategies (i.e., positive orientation, working harder, and seeking advice and assistance) and negatively associated with avoidance. With the exception of task complexity, resources were associated with lower levels of emotional exhaustion. Although a positive orientation was negatively associated with emotional exhaustion, another active coping strategy, namely, working harder, was positively associated with emotional exhaustion. We discuss the several complexities predicted and found, and how COR may be used elaborate the exhaustion model.     

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.     

Statistical manifold as an affine space: A functional equation approach

A statistical manifold M_μ consists of positive functions f such that defines a probability measure. In order to define an atlas on the manifold, it is viewed as an affine space associated with a subspace of the Orlicz space L^Φ. This leads to a functional equation whose solution, after imposing the linearity constrain in line with the vector space assumption, gives rise to a general form of mappings between the affine probability manifold and the vector (Orlicz) space. These results generalize the exponential statistical manifold and clarify some foundational issues in non-parametric information geometry.     

Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences

We re-examine the theoretical status of Fechner's Mathematical Auxiliary Principle [Fechner, G. T. (1889). Elemente der psychophysik. Leipzig: Breitkopf und Härtel] which underlies Fechner's method of constructing a sensory scale by integrating just noticeable differences (jnds). That “Principle” has been roundly criticized [Luce, R. D., & Edwards, W. (1958). The derivation of subjective scales from just noticeable differences. Psychological Review, 65, 227-237] as being inconsistent with the very basis of Fechner's psychophysical theory, and indeed this is the case. In important papers Pfanzagl [(1962). Über die stochastische Fundierung des psychophysischen Gesetzes (On the stochastic foundations of the psychophysical law). Biometrische Zeitschrift, 4, 1-14] and Krantz [(1971). Integration of just noticeable differences. Journal of Mathematical Psychology, 8, 591-599] resurrected Fechner's method; their analysis showed that the sensory scale could be written as the limit of a sequence of integrals, each of the form suggested by the auxiliary principle. In this work, we investigate the properties of a typical member of the Krantz-Pfanzagl sequence of integrals; we do so with the view to obtaining useful approximations to the true scale. Weber's inequality [Falmagne, J.-Cl. (1977). Note: Weber's inequality and Fechner's problem. Journal of Mathematical Psychology, 16, 267-271] plays an important role in our developments. That inequality, and other inequalities of a similar nature, allows us to place bounds on the error incurred by approximating a true scale u by an integral of jnds. Under appropriate conditions these bounds are sufficiently tight that the relative error is very small over the entire stimulus domain. We illustrate our theoretical results with a number of examples.     

Binary choice, subset choice, random utility, and ranking: A unified perspective using the permutahedron

The d-permutahedron Π_d−1⊂R^d is defined as the convex hull of all d-dimensional permutation vectors, namely, vectors whose components are distinct values of a d-element set of integers [d]≡{1,2,…,d}. By construction, Π_d−1 is a convex polytope with d! vertices, each representing a linear order (ranking) on [d], and has dimension dim(Π_d−1)=d−1. This paper provides a review of some well-known properties of a permutahedron, applies the geometric-combinatoric insights to the investigation of the various popular choice paradigms and models by emphasizing their inter-connections, and presents a few new results along this line.Permutahedron provides a natural representation of ranking probability; in fact it is shown here to be the space of all Borda scores on ranking probabilities (also called “voters profiles” in the social choice literature). The following relations are immediate consequences of this identification. First, as all d! vertices of Π_d−1 are equidistant to its barycenter, Π_d−1 is circumscribed by a sphere S^d−2 in a (d−1)-dimensional space, with each spherical point representing an equivalent class of vectors whose components are defined on an interval scale. This property provides a natural expression of the random utility model of ranking probabilities, including the condition of Block and Marschak. Second, Π_d−1 can be realized as the image of an affine projection from the unit cube of dimension d(d−1)/2. As the latter is the space of all binary choice vectors describing probabilities of pairwise comparisons within d objects, Borda scores can be defined on binary choice probabilities through this projective mapping. The result is the Young's formula, now applicable to any arbitrary binary choice vector. Third, Π_d−1 can be realized as a “monotone path polytope” as induced from the lift-up of the projection of the cube onto the line segment [0,d]⊂R¹. As the 2^d vertices of the d-cube are in one-to-one correspondence to all subsets of [d], a connection between the subset choice paradigm and ranking probability is established. Specifically, it is shown here that, in the case of approval voting (AV) with the standard tally procedure (Amer. Pol. Sci. Rev. 72 (1978) 831), under the assumption that the choice of a subset indicates an approval (with equal probability) of all linear orders consistent with that chosen subset, the Brams-Fishburn score is then equivalent to the Borda score on the induced profile. Requiring this induced profile (ranking probability) to be also consistent with the size-independent model of subset choice (J. Math. Psychol. 40 (1996) 15) defines the “core” of the AV Polytope. Finally, Π_d−1 can be realized as a canonical projection from the so-called Birkhoff polytope, the space of rank-position probabilities arising out of the rank-matching paradigm; thus Borda scores can be defined on rank-position probabilities. To summarize, the many realizations of a permutahedron afford a unified framework for describing and relating various ranking and choice paradigms.     

Representation of subjective preferences under ambiguity

The objective of this paper is to show how potentially incomplete preferences of a decision maker (DM) on acts can be modelled formally in a subjective ambiguity perspective. We identify acts as functions from a state space Ω to bounded support (finitely additive) probabilities over a set X of prizes. Then, we characterize preferences over equibounded acts a which have a numerical representation by the family of functionals , where u is a cardinal utility on X (representing the risk attitude of the DM) and Π is a unique pointwise closed convex set of probabilities on all events in Ω (representing the ambiguity perceived by the DM). To this end, in addition to the usual independence and continuity assumptions, we add completeness and dominance for preferences restricted to constant acts; moreover, we consider two other properties (subjective monotonicity and coherence) related with the preferences of a DM who is not able, owing to his partial knowledge, to evaluate any event in Ω.