Learning Coefficients in Semi-Regular Models

Statistical learning models such as multilayer neural networks and mixed distributions are widely used, and understanding the accuracy of these models is crucial for their use. Recent advances have clarified theoretical learning accuracy in Bayesian inference, where metrics such as generalization loss and free energy are used to measure the accuracy of predictive distributions. It has become clear that the asymptotic behavior of these metrics is determined by a rational number specific to each statistical model, known as the learning coefficient (real log canonical threshold). The problem of determining the learning coefficient is known to be reducible to the problem of finding the normal crossing of Kullback-Leibler divergence in relation to algebraic geometry. In this context, it is crucial to perform appropriate coordinate transformations and blow-ups. This paper attempts to derive appropriate variable transformations and blow-ups from the properties of the log-likelihood ratio function. That is, instead of dealing with the Kullback-Leibler information itself, it uses the properties of the log-likelihood ratio function before taking the expectation to calculate the real log canonical threshold. This approach has not been considered in previous research. Using these variable transformations and blow-ups, this paper provides the exact values of the learning coefficients and their calculation methods for statistical models that meet simple conditions next to the regular conditions (referred to as semi-regular models), and as specific examples, provides the learning coefficients for semi-regular models with two parameters and for those models where the random variables take a finite number of values.