When trying to compute variational bounds (as derived in the previous post), a naive attempt to approximate the involved expectations (e.g. using a Taylor expansion) may destroy the bound.
This is where the Higher-order Jensen-Feynman inequality comes in. It allows us to do a higher-order polynomial expansion without destroying the variational bound.
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.