Thurstonian-type representations for "same-different" discriminations: probabilistic decisions and interdependent images

• Authors: Ehtibar N. Dzhafarov
• Venue: Journal of Mathematical Psychology
• Year: 2003

A general Thurstonian-type representation (with stochastically interdependent images and probabilistic decisions) for a "same-different" discrimination probability function Φ(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 Φ(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".