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 psi(x, y) is a model in which the two stimuli x, y are mapped into two generally interdependent random images P(x) and Q(y) 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 x and y in this trial are judged to be different. While stochastically interdependent, P(x) and Q(y) are selectively attributed to (influenced by), respectively, x and y, which is understood as the possibility of conditioning P(x) and Q(y) on some random variable R that renders them stochastically independent, with their conditional distributions selectively depending on, respectively, x and y. A general Thurstonian-type representation is considered "well-behaved" if the conditional probability with which P(x) and Q(y), 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 x and y. It is shown that no such well-behaved Thurstonian-type representation can account for psi(x, y) 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". (C) 2003 Elsevier Science (USA). All rights reserved.