  Non-linear and non-Gaussian signal inference problems are difficult to
tackle. Renormalization techniques permit us to construct good estimators for
the posterior signal mean within information field theory (IFT), but the
approximations and assumptions made are not very obvious. Here we introduce the
simple concept of minimal Gibbs free energy to IFT, and show that previous
renormalization results emerge naturally. They can be understood as being the
Gaussian approximation to the full posterior probability, which has maximal
cross information with it. We derive optimized estimators for three
applications, to illustrate the usage of the framework: (i) reconstruction of a
log-normal signal from Poissonian data with background counts and point spread
function, as it is needed for gamma ray astronomy and for cosmography using
photometric galaxy redshifts, (ii) inference of a Gaussian signal with unknown
spectrum and (iii) inference of a Poissonian log-normal signal with unknown
spectrum, the combination of (i) and (ii). Finally we explain how Gaussian
knowledge states constructed by the minimal Gibbs free energy principle at
different temperatures can be combined into a more accurate surrogate of the
non-Gaussian posterior.
