Mean Field Variational Bayes for Elaborate Distributions

Abstract. We develop strategies for mean field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely define elaborate distributions to be those having more complicated forms compared with common distributions such as those in the Normal and Gamma families. Examples are Asymmetric Laplace, Skew Normal and Generalized Ex-treme Value distributions. Such models suffer from the difficulty that the param-eter updates do not admit closed form solutions. We circumvent this problem through a combination of (a) specially tailored auxiliary variables, (b) univariate quadrature schemes and (c) finite mixture approximations of troublesome den-sity functions. An accuracy assessment is conducted and the new methodology is illustrated in an application