After reading this article you will learn about the applications of mathematical models to probability learning.
Application of mathematical equations and formulae has become a hallmark of any science and psychology is no exception. Starting from the early psychological experiments of Weber and Fechner there has been a steady rise in this trend in psychology towards quantification.
While some people may wonder whether complex human behaviour can be quantified and translated into mathematical language, nevertheless the growing tendency to apply the findings of psychology to situations and which involves predictions of behaviour has resulted in the development of mathematical models, to examine and understand different categories of human behaviour, and learning is no exception.
In fact, a pioneer in the application of mathematics to learning was Clark Hull. More recently, attempts have been made to apply mathematics, particularly the concept of probability to explain the learning process. It is hoped that the reader is aware of the concept of probability.
In recent years psychologists, the most prominent among them is Estes, have attempted to apply the concept of probability to the learning process. Estes proceeded on the following premise. When an individual begins to learn a new task, there are several possible responses available to him including the correct response.
Assuming that there is only one correct response R, the probability of this response occurring in the first attempt itself, is equal to the likelihood of all possible responses which are not correct responses. This means that the correct response occurring (or for that matter of any of the response is 1/NR, where NR stands for the total number of possible non-responses.)
If the total number of probable responses is 4, then 1/NR = 1/4 or 0.25. Gradually as the learning proceeds, the frequency of occurrence or the probability of occurrence of R gradually increases because of reinforcement of R and non-reinforcement of R and extinction of the other responses, the probability of R occurring increases and those of the other decreases.
When the learning becomes complete or perfect, the other responses are not likely to occur, and the probability of each of them often becomes zero or very nearly so. At the same time, the probability of occurrence of R reaches 1 or 100%.
In practice, this may not be exactly the case always and there may be exceptions. But for all practical purposes, we may assume this. The mathematical models assume that ‘learning’ is an event and that the job of mathematical or statistical theory or model is to predict the occurrences of the current response, at different stages and under different conditions.
The mathematical theories of learning are not concerned about the ‘what and how and why’ of learning, but about the ‘how much’ of it. These ‘learned responses’ are a set of events with different degrees of probability of occurrence under different conditions.
The formulations of Estes are based on these assumptions. Estes arrived at a set of equations, which would help us to predict the probability of occurrence of the ‘right’ response under different conditions. It is not possible to discuss or explain these in detail here.
However some of the assumptions made by Estes may be stated here:
(a) According to Estes responses or reactions of organisms can be classified into two broad categories, those that result in a particular ‘end act’ or outcome (A1) and those that do not (A2). Thus, the actions of Skinner’s rat or Thorndike’s cat could end either in the former pressing the bar or not and the latter pulling the rope or not.
(b) Similarly on the stimulus side also one can assume the real or operative elements in the situation can vary in the probability of their occurrence.
(c) Estes basing his model on the process of conditioning, and S-R learning as the basic model states, at the start of the learning process, the Al response, the one resulting in an outcome, and A2 responses that do not lead the outcome, are all not conditioned to associate with any stimulus element.
Similarly, A1 and A2 responses get associated with different stimulus elements. Initially A1 and A2 responses get associated with different stimulus elements. Gradually the positive or Al response gets conditioned to or associated with more and more elements in the stimulus situation and these S-R bonds get strengthened. Thus, there is a steady increase in the probability of occurrence of elements in the stimulus situation getting associated with the positive response A1. As reinforcement continues with the positive response A1, the strength of the connection between this response and different elements increases.
Based on this assumption, Estes arrived at a series of equations which would help us to predict the probability occurrence of the right response under different conditions. It is not possible to explain these equations in detail here. The mathematical model and the applications of the concept of probability to the problem of learning certainly represents an advance in our ability to predict the occurrence of the correct response but the ideal of being able to predict is yet a very long way off.
These models have not proved yet very effective. Another criticism is that the mathematical theories have based their models entirely on the S-R model totally ignoring the role of cognitive processes and this is admitted by these theories themselves. These models do not attempt to explain the nature and course of the learning process but only predict the probability of occurrence of the correct response like a street comer palmist or soothsayer.