Free energy minimization: a unified framework for modeling, inference, learning, and optimization [lecture notes]

The goal of this lecture note is to review the problem of free energy minimization as a unified framework underlying the definition of maximum entropy modeling, generalized Bayesian inference, learning with latent variables, the statistical learning analysis of generalization, and local optimization. Free energy minimization is first introduced, here and historically, as a thermodynamic principle. Then, it is described mathematically in the context of Fenchel duality. Finally, the applications to modeling, inference, learning, and optimization are covered, starting from basic principles.