Before revision: Hierarchical modeling and learning has proven very powerful in the field of Gaussian process regression and kernel methods, especially for machine learning applications and , increasingly, within the field of inverse problemsmore generally. The classical approach to learning hierarchical information is through Bayesian formulations of the problem, implying a posterior distribution on the hierarchical parameters or, in the case of empirical Bayes, providing an optimization criterion for them. Recent developments in the machine learning literature have suggested new criteria for hierarchical learning , based on approximation theoretic considerations that can be interpreted as variants of cross-validation, and exploiting approximation consistency in data splitting .
After revision: Hierarchical modeling and learning has proven very powerful in statistics, machine learning , based on approximation theoretic considerations that can be interpreted as variants of cross-validation, and exploiting approximation consistency in data splitting .