Evaluation of effective degree of freedom of physical model using learning coefficient

Abstract

In algebraic-geometric learning theory, it has been suggested that learning coefficients, which are statistics associated with the model evaluation of a singular model, can be applied to evaluate the effective degrees of freedom of the model. Such information can be very useful in constructing reduced-form models of complex physical phenomena. However, it is difficult to derive the learning coefficients analytically for many physical models. The Bayesian free energy, which represents the goodness of fit of the learning model to the data, and the learning coefficient are closely related. For complex models, the Bayesian free energy is often estimated using the exchange Monte Carlo method, a type of Monte Carlo sampling method. A method to estimate the learning coefficient from the exchange rate obtained during exchange Monte Carlo sampling has been proposed. In this study, we apply this method to a physical model for which it is difficult to calculate the analytical Bayesian free energy and verify that it is possible to estimate the effective degree of freedom of the target. In particular, we confirm that the effective degree of freedom can be estimated for the Sloppy model, which is related to the survival time model of biological systems.