By mathematical learning theory we mean to denote the growing body of research methods and results concerned with the conceptual representation of learning phenomena, the mathematical formulation of assumptions or hypotheses about learning, and the derivation of testable theorems. The application of mathematics to problems arising in the psychology of learning is not in itself a new development. Efforts toward quantification have been a persisting motif throughout the history of this discipline and have become particularly prominent in the influential contemporary theories of Hull and Spence. Still, the student of learning cannot but be impressed by the sharp increase, perhaps by a factor of 10 or 20 during the present decade, in the number of persons participating in these efforts, and by the appearance of researches devoted to the formulation and analysis of relatively general mathematical models which offer possibilities of diverse interpretations within the broad area of learning theory. This work on mathematical models for learning has not attempted to formalize any particular theoretical system of behavior; yet the influences of Guthrie and Hull are most noticeable. Compared with the older attempts at mathematical theorizing, the recent work has been more concerned with detailed analyses of data relevant to the models and with the design of experiments for directly testing quantitative predictions of the models.
Two main lines of inquiry have been followed. One has been concerned with the development of a model of the stimulus environment of an experimental subject and with the processes by which that environment influences behavior. In the literature, this research has been referred to as "statistical learning theory" or "stimulus-sampling models." The other main line of inquiry has been concerned with models of the time sequences of responses and with the detailed properties of the stochastic processes defined by those models. Such work is most commonly referred to by the name "stochastic learning models."
There is no basic conflict between these two avenues of research. In fact, they are complementary to one another. The stimulus-sampling models lead to theorems about how response probabilities are transformed from one trial to another. The stochastic response models are concerned with the consequences of such transformation rules. In addition, some of the stimulus- sampling models predict quantitative relations among the parameters that . . .