  Variational approaches to approximate Bayesian inference provide very
efficient means of performing parameter estimation and model selection. Among
these, so-called variational-Laplace or VL schemes rely on Gaussian
approximations to posterior densities on model parameters. In this note, we
review the main variants of VL approaches, that follow from considering
nonlinear models of continuous and/or categorical data. En passant, we also
derive a few novel theoretical results that complete the portfolio of existing
analyses of variational Bayesian approaches, including investigations of their
asymptotic convergence. We also suggest practical ways of extending existing VL
approaches to hierarchical generative models that include (e.g., precision)
hyperparameters.
