This post is going to cover Variational Bayesian methods and, in particular, the most common one, the mean-field approximation. This is a topic that I've been trying to understand for a while now but didn't quite have all the background that I needed. After picking up the main ideas from variational calculus and getting more fluent in manipulating probability statements like in my EM post, this variational Bayes stuff seems a lot easier.
Variational Bayesian methods are a set of techniques to approximate posterior distributions in Bayesian Inference. If this sounds a bit terse, keep reading! I hope to provide some intuition so that the big ideas are easy to understand (which they are), but of course we can't do that well unless we have a healthy dose of mathematics. For some of the background concepts, I'll try to refer you to good sources (including my own), which I find is the main blocker to understanding this subject (admittedly, the math can sometimes be a bit cryptic too). Enjoy!