The Graphical Test Theory from Bayesian Approach

Abstract

This paper proposes a new test theory, which is called the graphical test theory, from Bayesian approach. The unique features of this method are as follows ; 1. the joint probability distribution of test data is represented by the probabilistic network model, 2. the test construction algorithm by using the amount of test information is proposed from the decision theory, and 3. the predictive test score and predictive learner's knowledge states can be provided. From the mathematical view point, we assume the Dirichlet-multinomial model for the network model, which is more moderate assumption than one of the traditional modern test theories. From this, we derive a maximum a posterior (MAP) estimator of parameters and the exact solutions of the predictive distributions of the structures for the networks. Furthermore, we provide the following results : When the hyper-parameter is 1 (the prior is uniform), this model asymptotically becomes close to the Item Response Theory as the number of items becomes large. The amount of Test Information of this model becomes close to the amount of Fischer Information as the number of items becomes large. In addition, the estimation of a network structure given data in this theory is equivalent to one of the Item Relational Structure Analysis (IRS Analysis) when the hyper-parameter is 1/(2 log n) (the asymptotic likelihood form).