Robert R. Bush

A Mathematical Model for Simple Learning

A mathematical model for simple learning is presented. Changes in the probability of occurrence of a response in a small time h are described with the aid of mathematical operators. The parameters which appear in the operator equations are related to experimental variables such as the amount of reward and work. Relations between the probability and empirical measures of rate of responding and latent time are defined. Acquisition and extinction of behavior habits are discussed for the simple runway and for the Skinner box. Equations of mean latent time as a function of trial number are derived for the runway problem; equations for the mean rate of responding and cumulative numbers of responses versus time are derived for the Skinner box experiments. An attempt is made to analyze the learning process with various schedules of partial reinforcement in the Skinner type experiment. Wherever possible, the correspondence between the present model and the work of Estes [2] is pointed out.

A Comparison of Eight Models

In the testing of a scientific model or theory, one rarely has a general measure of goodness-of-fit, a universal yardstick by which one accepts or rejects the model. Indeed, science does not and should not work this way; a theory is kept until a better one is found. One way that science does work is by comparing two or more theories to determine their relative merits in handling relevant data. In this paper we present a comparison of eight models for learning by using each to analyze the data from the same experiment.1 A primary goal of any learning model is to predict correctly the learning curve—proportions of correct responses versus trials. Almost any sensible model with two or three free parameters, however, can closely fit the curve, and so other criteria must be invoked when one is comparing several models. A criterion that has been used in recent years is the extent to which a model can reproduce the fine-grain structure of the response sequences. Many properties can be and have been invented for this purpose. Fourteen such properties are used in this paper. A summary index of how well one model fits the fine-grain detail of data compared with another model is the likelihood ratio. There are three objections to this measure, however. First, for many models it is very difficult to compute. Second, its use obscures the particular strengths and weaknesses of a model and so fails to suggest why the model is inadequate. Third, it may be especially sensitive to uninteresting differences between the model and the experiment. Therefore we do not use likelihood ratios in this paper. A satisfactory prediction of the sequential properties of learning data from a single experiment is by no means a final test of a model. Numerous other criteria—and some more demanding—can be specified. For example, a model