Equations of States in Statistical Learning for a Nonparametrizable and Regular Case

Many learning machines that have hierarchical structure or hidden variables are now being used in information science, artificial intelligence, and bioinformatics. However, several learning machines used in such fields are not regular but singular statistical models, hence their generalization performance is still left unknown. To overcome these problems, in the previous papers, we proved new equations in statistical learning, by which we can estimate the Bayes generalization loss from the Bayes training loss and the functional variance, on the condition that the true distribution is a singularity contained in a learning machine. In this paper, we prove that the same equations hold even if a true distribution is not contained in a parametric model. Also we prove that, the proposed equations in a regular case are asymptotically equivalent to the Takeuchi information criterion. Therefore, the proposed equations are always applicable without any condition on the unknown true distribution

Statistical Learning Theory of Quasi-Regular Cases

Many learning machines such as normal mixtures and layered neural networks are not regular but singular statistical models, because the map from a parameter to a probability distribution is not one-to-one. The conventional statistical asymptotic theory can not be applied to such learning machines because the likelihood function can not be approximated by any normal distribution. Recently, new statistical theory has been established based on algebraic geometry and it was clarified that the generalization and training errors are determined by two birational invariants, the real log canonical threshold and the singular fluctuation. However, their concrete values are left unknown. In the present paper, we propose a new concept, a quasi-regular case in statistical learning theory. A quasi-regular case is not a regular case but a singular case, however, it has the same property as a regular case. In fact, we prove that, in a quasi-regular case, two birational invariants are equal to each other, resulting that the symmetry of the generalization and training errors holds. Moreover, the concrete values of two birational invariants are explicitly obtained, the quasi-regular case is useful to study statistical learning theory