Real Log Canonical Thresholds at Non-singular Points

TL;DR

This paper introduces a method to compute the real log canonical threshold at many non-singular points in statistical models, enabling better bounds on learning coefficients, especially for singular models where exact values are hard to determine.

Contribution

It extends previous work by providing a formula for calculating the real log canonical threshold at many non-singular points, facilitating bounds on learning coefficients in complex models.

Findings

Provides a formula for many non-singular points in realizable parameters.

Enables upper bounds for learning coefficients in singular models.

Confirms the method's validity with examples including mixed binomial and reduced-rank regression models.

Abstract

Recent advances have clarified theoretical learning accuracy in Bayesian inference, revealing that the asymptotic behavior of metrics such as generalization loss and free energy, assessing predictive accuracy, is dictated by a rational number unique to each statistical model, termed the learning coefficient (real log canonical threshold). For models meeting regularity conditions, their learning coefficients are known. However, for singular models not meeting these conditions, exact values of learning coefficients are provided for specific models like reduced-rank regression, but a broadly applicable calculation method for these learning coefficients in singular models remains elusive. This paper extends the application range of the previous work and provides an approach that can be applied to many points within the set of realizable parameters. Specifically, it provides a formula for calculating the real log canonical threshold at many non-singular points within the set of realizable parameters. If this calculation can be performed, it is possible to obtain an upper bound for the learning coefficient of the statistical model. Thus, this approach can also be used to easily obtain an upper bound for the learning coefficients of statistical models. As an application example, it provides an upper bound for the learning coefficient of a mixed binomial model, and calculates the learning coefficient for a specific case of reduced-rank regression, confirming that the results are consistent with previous research.