  Cognitive diagnostic assessment aims to measure specific knowledge structures
in students. To model data arising from such assessments, cognitive diagnostic
models with discrete latent variables have gained popularity in educational and
behavioral sciences. In a learning context, the latent variables often denote
sequentially acquired skill attributes, which is often modeled by the so-called
attribute hierarchy method. One drawback of the traditional attribute hierarchy
method is that its parameter complexity varies substantially with the
hierarchy's graph structure, lacking statistical parsimony. Additionally,
arrows among the attributes do not carry an interpretation of statistical
dependence. Motivated by these, we propose a new family of latent conjunctive
Bayesian networks (LCBNs), which rigorously unify the attribute hierarchy
method for sequential skill mastery and the Bayesian network model in
statistical machine learning. In an LCBN, the latent graph not only retains the
hard constraints on skill prerequisites as an attribute hierarchy, but also
encodes nice conditional independence interpretation as a Bayesian network.
LCBNs are identifiable, interpretable, and parsimonious statistical tools to
diagnose students' cognitive abilities from assessment data. We propose an
efficient two-step EM algorithm for structure learning and parameter estimation
in LCBNs. Application of our method to an international educational assessment
dataset gives interpretable findings of cognitive diagnosis.
