Abstract
In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde matrix-type singularities, we show the efficiency of such methods.
Highlights
In recent studies, real data associated with, for example, image or speech recognition, psychology, and economics, have been analyzed by learning systems
Consider a learning model that is written in probabilistic form as p( x |w), where w ∈ W ⊂ Rd is a parameter
The remainder of this paper is structured as follows: In Section 2, we introduce log canonical thresholds in algebraic geometry
Summary
Real data associated with, for example, image or speech recognition, psychology, and economics, have been analyzed by learning systems. The WAIC and the cross-validation can estimate the Bayesian generalization error without any knowledge of the true probability density function These values are calculated from training samples xi using learning model p. We consider the value λ, which is equal to the log canonical threshold introduced in Definition 1 This coefficient is not needed to evaluate the WAIC and the cross-validation in practice, while the learning coefficients from our recent results have been used very effectively by Drton and Plummer [8] for model selection using a method called the “singular Bayesian information criterion (sBIC)”. From learning coefficient λ and its order θ, value ν is obtained theoretically as follows: Let ξ (u) be an empirical process defined on the manifold obtained by a resolution of singularities and.
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