A Latent Trait Theory Via a Stochastic Learning Theory for a Knowledge Space

To capture the cognitive organization of a set of questions or problems pertaining to a body of information, Doignon and Falmagne have proposed, and analyzed in a number of papers, the concept of a knowledge space , that is, a distinguished collection of subsets of questions, representing the possible knowledge states . This collection of sets is assumed to satisfy a number of conditions. Since this concept is a deterministic one, the problem of empirical testing arises. A stochastic version of a knowledge space is developed in this paper, in which the knowledge states are considered as possible epochs in a subject's learning history. The knowledge space is decomposed as a union of a number of possible learning paths, called gradations . The model specifies how a subject is channelled through and progresses along a gradation. A probabilistic axiom of the “local indepencence” type relates the knowledge states to the observable responses. The predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions. An application of the model to artificial data is described, based on maximum likelihood methods. The statistical analysis is shown to be capable of revealing the combinatoric core of the model.