The relative entropy is frequently framed as the inverse temperature times the difference between the Helmholtz and variational/Bethe free energies, the latter of which is minimized as a proxy objective function in mean field theory and Bayesian estimation via the well-known expectation maximization algorithm.

The relative entropy is frequently framed as the inverse temperature times the difference between the Helmholtz and variational/Bethe free energies, the latter of which is minimized as a proxy objective function in mean field theory and Bayesian estimation via the well-known expectation maximization algorithm. MacKay's book is a good reference for this.