By now, I hope all machine learners are convinced of the importance of variational methods for approximated inference and learning in general. Specially given the fast increase in popularity of those methods (NIPS15, NIPS14).
As a follow up of my posts on partition functions ( part1, part2 and part3 ), I was inspired by a couple of papers this last NIPS ( paper1, paper2 ) to expand/review a little more the methods for approximating partition functions and free energies in statistical mechanics.
Variational Inference is a technique which consists in bounding the log-likelihood ln p(x) defined by a model with latent variables p(x,z)=p(x|z)p(z) through the introduction of a variational distribution q(z|x) with same support as p(z):
Often the expectations in the bound F(x) (aka, ELBO or Free Energy) cannot be solved analytically.
In some cases, we can make use of a few handful inequalities which I quickly summarize below.
Some of these inequalities introduce new variational parameters. Those should be optimized jointly with all the other parameters to minimize the ELBO.