We give an overview of statistical models and likelihood, together with two of its variants: penalized and hierarchical likelihood. The Kullback-Leibler divergence is referred to repeatedly in the literature, for defining the misspecification risk of a model and for grounding the likelihood and the likelihood cross-validation, which can be used for choosing weights in penalized likelihood. Families of penalized likelihood and particular sieves estimators are shown to be equivalent. The similarity of these likelihoods with a posteriori distributions in a Bayesian approach is considered.
"Statistical models: Conventional, penalized and hierarchical likelihood." Statist. Surv. 3 1 - 17, 2009. https://doi.org/10.1214/08-SS039