This paper derives the variational free energy under the Laplace approximation, focusing on how additional model complexity induced by increasing the number of model parameters affects the free energy. The authors show that restricted maximum likelihood (ReML) can be adjusted to provide an approximation to the log-evidence for a particular model, making it useful for model selection and comparison. They demonstrate that ReML can be used for automatic model selection and relevance determination (ARD) in hierarchical models, where priors are selected principledly. The paper also links variational Bayes (VB), expectation maximisation (EM), and ReML by deriving them from basic principles, revealing their relationships and showing that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. The authors further discuss dynamic models and how they inform the regularization of free energy schemes like EM and ReML. The paper concludes with a brief demonstration of model selection using ReML and its adjusted free energy in a hierarchical linear model.This paper derives the variational free energy under the Laplace approximation, focusing on how additional model complexity induced by increasing the number of model parameters affects the free energy. The authors show that restricted maximum likelihood (ReML) can be adjusted to provide an approximation to the log-evidence for a particular model, making it useful for model selection and comparison. They demonstrate that ReML can be used for automatic model selection and relevance determination (ARD) in hierarchical models, where priors are selected principledly. The paper also links variational Bayes (VB), expectation maximisation (EM), and ReML by deriving them from basic principles, revealing their relationships and showing that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. The authors further discuss dynamic models and how they inform the regularization of free energy schemes like EM and ReML. The paper concludes with a brief demonstration of model selection using ReML and its adjusted free energy in a hierarchical linear model.