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

TL;DR

This paper extends equations in statistical learning to nonparametrizable and regular models, enabling estimation of generalization loss without assumptions on the true distribution, and relates them to the Takeuchi information criterion.

Contribution

It proves that the equations for estimating generalization loss are valid for both singular and regular models, regardless of true distribution inclusion.

Findings

Equations hold even if the true distribution is outside the model.

In regular cases, equations asymptotically match the Takeuchi information criterion.

Applicable to a wider class of models without distribution assumptions.

Abstract

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.