Applications of geometric complexity and the minimum description length principle in mathematical modeling of cognition

Several model complexity dimensions such as the number of parameters, the functional form of the model, and parameter interdependence have been proposed and investigated in the field of model selection. No theoretical understanding has been achieved. A theoretical understanding of model complexity is needed.;Model selection methods, such as AIC and BIC, have been advanced and widely used in practice of model selection. They have shown successful performance for linear and nested models or in large sample situations. However, when the models have same number of parameters but different functional forms, these methods no longer work well. A new model selection criterion, which is sensitive to the effect of the functional form of the model and would work in small sample situations is needed.;Geometric complexity is a measure of model complexity that Rissanen (1996) recently proposed as an improved formulation of the Minimum Description Length (MDL) principle. The validity of geometric complexity as a measure of a model's data fitting capability is investigated. In three areas of cognitive modeling (psychophysics, information integration, categorization), two competing models with the same number of parameters but different functional form were compared in terms of geometric complexity. The model which showed superior data fitting capability turn out to have greater geometric complexity than the other competing model. This proves the validity of geometric complexity as a measure of a model's inherent ability of fitting diverse data patterns.;The application of geometric complexity as a model selection method is examined. Two competing models in the three areas of cognition were fit to data sets generated by each model. The model recovery rates for four selection methods (MDL with the geometric complexity and AIC, BIC, and ML) were compared. The results showed that MDL was superior to the other methods in recovering the model that generated the data. (Abstract shortened by UMI.).