Inference with minimal Gibbs free energy in information field theory

TL;DR

This paper introduces the minimal Gibbs free energy concept to information field theory, providing a unified framework for constructing estimators in complex non-linear, non-Gaussian inference problems, with practical applications in astrophysics and cosmology.

Contribution

It presents a novel approach using minimal Gibbs free energy to derive estimators in IFT, unifying previous renormalization results and enabling improved inference in challenging scenarios.

Findings

Unified framework for non-Gaussian inference

Optimized estimators for astrophysical signals

Combining Gaussian states for better posterior approximation

Abstract

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.