  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. The problem of determining learning coefficients
relates to finding normal crossings of Kullback-Leibler divergence in algebraic
geometry. In this context, it is crucial to perform appropriate coordinate
transformations and blow-ups. This paper introduces an approach that utilizes
properties of the log-likelihood ratio function for constructing specific
variable transformations and blow-ups to uniformly calculate the real log
canonical threshold. It was found that linear independence in the differential
structure of the log-likelihood ratio function significantly influences the
real log canonical threshold. This approach has not been considered in previous
research. In this approach, the paper presents cases and methods for
calculating the exact values of learning coefficients in statistical models
that satisfy a simple condition next to the regularity conditions (semi-regular
models), offering examples of learning coefficients for two-parameter
semi-regular models and mixture distribution models with a constant mixing
ratio.
