The two kinds of free energy and the Bayesian revolution
Fig 3
The normalization of a functon ϕ to obtain a probability distribution pϕ is equivalent to fitting trial distributions q to the shape of ϕ by minimizing free energy.
In two dimensions, the normalization of a point ϕ = (ϕ1, ϕ2) corresponds to a (non-orthogonal) projection onto the plane of probability vectors (A). For continuous domains, where probability distributions are represented by densities, normalization corresponds to a rescaling of ϕ such that the area below the graph equals 1 (B). Instead, when minimizing variational free energy (red colour), the trial distributions q are varied until they fit to the shape of the unnormalized function ϕ (perfectly at q = pϕ).