Mean field variational Bayes for continuous sparse signal shrinkage: Pitfalls and remedies

We investigate mean field variational approximate Bayesian inference for models that use continuous distributions, Horseshoe. Negative-Exponential-Gamma and Generalized Double Pareto, for sparse signal shrinkage. Our principal finding is that the most natural, and simplest, mean field variational Bayes algorithm can perform quite poorly due to posterior dependence among auxiliary variables. More sophisticated algorithms, based on special functions, are shown to be superior. Continued fraction approximations via Lentz's Algorithm are developed to make the algorithms practical.