  Variational Bayes methods approximate the posterior density by a family of
tractable distributions whose parameters are estimated by optimisation.
Variational approximation is useful when exact inference is intractable or very
costly. Our article develops a flexible variational approximation based on a
copula of a mixture, which is implemented by combining boosting, natural
gradient, and a variance reduction method. The efficacy of the approach is
illustrated by using simulated and real datasets to approximate multimodal,
skewed and heavy-tailed posterior distributions, including an application to
Bayesian deep feedforward neural network regression models.
