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.
Great! The piecewise linear/quadratic bound for \log (1+\exp(x)) here could also come in handy http://www.icml-2011.org/papers/376_icmlpaper.pdf
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