"I Can Name that Bayesian Network in Two Matrixes!"

Abstract For a number of situations, a Bayesian network can be split into a core network consisting of a set of latent variables describing the status of a system, and a set of fragments relating the status variables to observable evidence that could be collected about the system state. This situation arises frequently in educational testing, where the status variables represent the student proficiency and the evidence models (graph fragments linking competency variables to observable outcomes) relate to assessment tasks that can be used to assess that proficiency. The traditional approach to knowledge engineering in this situation would be to maintain a library of fragments, where the graphical structure is specified using a graphical editor and then the probabilities are entered using a separate spreadsheet for each node. If many evidence model fragments employ the same design pattern, a lot of repetitive data entry is required. As the parameter values that determine the strength of the evidence can be buried on interior screens of an interface, it can be difficult for a design team to get an impression of the total evidence provided by a collection of evidence models for the system variables, and to identify holes in the data collection scheme. A Q -matrix – an incidence matrix whose rows represent observable outcomes from assessment tasks and whose columns represent competency variables – provides the graphical structure of the evidence models. The Q -matrix can be augmented to provide details of relationship strengths and provide a high level overview of the kind of evidence available. The relationships among the status variables can be represented with an inverse covariance matrix; this is particularly useful in models from the social sciences as often the domain experts’ knowledge about the system states comes from factor analyses and similar procedures that naturally produce covariance matrixes. The representation of the model using matrixes means that the bulk of the specification work can be done using a desktop spreadsheet program and does not require specialized software, facilitating collaboration with external experts. The design idea is illustrated with some examples from prior assessment design projects.