Identifiability in probabilistic knowledge structures

The so-called basic local independence model (BLIM) constitutes the standard probabilistic model within the theory of knowledge structures. The present paper characterizes local identifiability of the BLIM through the rank of its Jacobian matrix. Within this framework, it reconsiders conditions known to give rise to non-identifiability, and presents some new cases. Together they completely cover the instances cropping up in the collection of BLIMs arising from all the possible knowledge structures on a three-item domain. The derived theoretical results, providing a full account of the trade-offs between parameters that occur in these situations, hold for arbitrary BLIMs, and are not limited to domains of particular cardinality. Moreover, it is shown that previously formulated conjectures on the encountered types of parameter trade-offs need not hold on the whole parameter space, but in general may only be true almost everywhere. If this indeed is the case, remains an open problem.