BLIM’s identifiability and parameter invariance under backward and forward transformations

Abstract

The basic local independence model (BLIM) is one of the most widely applied probabilistic models in knowledge space theory. It is known that the BLIM is not identifiable in general and that its identifiability strictly depends on the properties of the knowledge structure to which it is applied. If the knowledge structure is either forward- or backward-graded in one or more items, then the BLIM is not identifiable. In such cases, there exist continuous transformations of the model’s parameters, named forward and backward transformations, that keep constant the value of the model’s prediction function. Under certain constraints on the model’s parameters, some of the transformations might lose this property. The type of constraints considered in this article consist of fixing the probability of a knowledge state to a constant value. The theoretical results contained in the article shed light on the role of the different state probabilities in reducing the collection of transformations and thus restoring the identifiability of the model’s parameters.