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

Flexible Variational Bayes based on a Copula of a Mixture