Some considerations on the factorization of state probabilities in knowledge structures

In most of the work on probabilistic knowledge structures, no restrictions apply to the probability distribution on the collection of knowledge states. The number of parameters needed to describe this distribution can thus be very large. An exception is provided by the Simple Learning Model (SLM) in which, by means of assumptions on the learning process, the distribution is built from other parameters. However, the application of the SLM is limited to learning spaces. The present work generalizes the SLM by suggesting a method to build the state probabilities as products of the probabilities of single (or groups of) items. The construction follows from first factorizing the probability distribution into the product of marginal and conditional probabilities based on the blocks of an ordered exact cover of the item domain, and then by imposing suitable constraints on these probabilities. A specific ordered exact cover of the item domain, distinguishing minimal from non-minimal items, is shown to be of particular interest as it allows to recover the SLM and to generalize it to a wider class of regular knowledge structures.