Estes from Stimulus - Sampling to Array - Similarity Theory

The development of mathematical models of cognition from 1950 to the 1980s was shaped to an important degree by efforts to account for three empirical phenomena probability matching in simple multiple-choice learning, generalization as a function of commonality of learning and test situations, and the learning of perfect discriminations between similar stimuli. Among the models of the 1 9 5 0 ~ ~ linear models accounted for the first of these and stimulus sampling theory for the first two, but the third remained elusive for more than two decades. The third of the phenomena was brought into the fold with the emergence of array-similarity theory in the late 1970s. The critical change from stimulus sampling theory was the replacement of an additive similarity function with a multiplicative function, the product rule. Analyses within the general framework of the work of Estes and Suppes on foundations of linear and stimulus-sampling models shows that the stimulus-sampling and arraysimilarity models are isomorphic in some very special cases, exhibit close agreement in their predictions whenever stimulus structures do not allow significant nonlinearity of similarity measures, but in many cases, especially those involving discrimination or categorization of multidimensional stimuli, may diverge widely. The nonlinearities in the array-similarity model make prohibitively difficult the derivation of functions like the learning curves and descriptives statistics of data that were the bread-and-butter products of linear and stimulus-sampling models, but application and testing of both types of models can now be readily accomplished via implementation of the models on high speed computers. Three basic facts about learning have been cornerstones in the development of mathematical learning theory: (1) probability matching; (2) transfer of learning in direct relation to the commonality of training and test situations; and (3) discrimination between similar stimuli. The first two of these were treated with considerable success by the family of learning models on which Patrick Suppes and I worked jointly in the 1950s, but the goal of handling all three within a unified theory eluded us. There would have been no difficulty if we had been willing to augment our theory with ad hoc mechanisms, but we persisted in feeling that a parsimonious solution could be achieved by some revision of our core assumptions. I think now that we were right, and in this paper I Fi Hurnphreys (ed.), Patrick Suppes: Scientific Philosop/pher; Vol. 3, 171-192. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.