Probabilistic Knowledge Structures

Abstract

The concept of a knowledge structure is a deterministic one. As such, it does not provide realistic predictions of subjects’ responses to the problems of a test. There are two ways in which probabilities must enter in a realistic model. For one, the knowledge states will certainly occur with different frequencies in the population of reference. It is thus reasonable to postulate the existence of a probability distribution on the collection of states. For another, a subject’s knowledge state does not necessarily specify the observed responses. A subject having mastered an item may be careless in responding, and make an error. Also, in some situations, a subject may be able to guess the correct response to a question not yet mastered. In general, it makes sense to introduce conditional probabilities of responses, given the states. A number of simple probabilistic models will be described in this chapter. They will be used to illustrate how probabilistic concepts can be introduced within knowledge space theory. These models will also provide a precise context for the discussion of some technical issues related to parameter estimation and statistical testing. In general, this material must be regarded as a preparation for the stochastic theories discussed in Chapters 8, 10 and 11.