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Building adaptive tests using Bayesian networks

Jirí Vomlel

Abstract

1 JlŘÍ VOMLEL Wc propose a framework for building decision strategies using Bayesian network models and discuss its application to adaptive testing. Dynamic programming and AO* algorithm are used to find optimal adaptive tests. The proposed AO* algorithm is based on a new admissible heuristic function. 1. BAYESIAN NETWORKS Bayesian networks are probabilistic graphical models that are capable of modelling domains comprising uncertainty. They were introduced to the field of expert systems by Pearl [17] and Spiegelhalter and R. P. Knill-Jones [21]. The first applications were an expert system for electromyography Munin [2] and the Pathfinder system [3]. Since then Bayesian networks were successfully applied in several areas. Strength of graphical models is not only that they enable efficient uncertainty reasoning with hundreds of variables (e.g. using the method of Lauritzen and Spiegelhalter [13]), but also they help humans to understand better the modelled domain. This is mainly due to their comprehensible representation by use of directed acyclic graphs representing dependencies between domain variables. See [14] where some recent applications of Bayesian networks are discussed. Bayesian network consists of an directed acyclic graph (DAG) G = (V,E), to each node i G V corresponds one random variable Xi with a finite set X; of mutually exclusive states and a conditional probability table (CPT) P(Xi \ (Xj)j epa ^), where pa(i) denotes the set of parents of node i in graph G. See Figure 1 for an example of Bayesian network. Bayesian network encodes qualitative and quantitative knowledge. Quantitative knowledge is represented by CPTs, while qualitative is encoded by use of a DAG. The DAG implies certain conditional independence relations between variables (Xi)i^v

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