Learning coefficients for hierarchical learning models in Bayesian Estimation

Recently, singular learning theory has been analyzed using algebraic geometry as its basis. It is essential to determine the normal crossing divisors of learning machine singularities through a blowing-up process to observe the behaviors of state probability functions in learning theory. In this paper, we investigate learning coefficients for multi-layered neural networks with linear units, especially when dealing with a large number of layers in Bayesian estimation. We make use of the valuable results obtained by Aoyagi(2023), which provide the main terms for Bayesian generalization error and the average stochastic complexity (free energy). These terms are widely employed in numerical experiments, such as in information criteria