  We derive and present explicit algorithms to facilitate streamlined computing
for variational inference for models containing higher level random effects.
Existing literature, such as Lee and Wand (2016), is such that streamlined
variational inference is restricted to mean field variational Bayes algorithms
for two-level random effects models. Here we provide the following extensions:
(1) explicit Gaussian response mean field variational Bayes algorithms for
three-level models, (2) explicit algorithms for the alternative variational
message passing approach in the case of two-level and three-level models, and
(3) an explanation of how arbitrarily high levels of nesting can be handled
based on the recently published matrix algebraic results of the authors. A
pay-off from (2) is simple extension to non-Gaussian response models. In
summary, we remove barriers for streamlining variational inference algorithms
based on either the mean field variational Bayes approach or the variational
message passing approach when higher level random effects are present.
