Invited Speaker---Dr. Miki Aoyagi, Associate Professor
Department of Mathematics, College of Science & Technology, Nihon University, Japan
Miki Aoyagi received his B.E., M.E., and D.E. degrees from Kyusyu University in 1994, 1995 and 1997, respectively by using a system which permits skipping grades. Since then she was working as PD Research Fellow in Kyusyu University, Hiroshima University, Hamburg University and The University of Tokyo until 2001. Currently she is an Associate Professor at Department of Mathematics, College of Science & Technology, Nihon University. Her research interests include learning theory, learning coefficients, resolution of singularities in algebraic geometry and information criteria.
Learning Coefficients and Information Criteria
In recent studies, image or speech recognition, psychology, and economics, etc. in real big data have been analyzed by learning systems. It is one of important problems to approximate an unknown true density function from training data selected from the true density function independently and identically using learning models. In a stochastic model, many hierarchical learning models for analysing real data have been proposed, and proved to be effective. They are, however, singular, which classic theories for regular models cannot apply to. Therefore, the need for appropriate model selection methods for singular models has increased and several information criteria for singular models have been developed. For example, singular Bayesian information criterion, widely applicable information criterion, widely applicable Bayesian information criterion, and cross-validation have been considered based on mathematical theorems in algebraic analysis and geometry. In this paper, we consider learning coefficients in learning theory, which serve to measure the main term of learning efficiency in singular learning models. These coefficients have an important role in information criteria and are mathematically equal to the log canonical thresholds of Kullback functions. We show several mathematical theorems for obtaining these coefficients and apply these theorems to Poisson distribution mixture models.