A review of Information Field Theory for Bayesian inference of random fields

Several physical problems require Bayesian inference of spatial, or spatio-temporal phenomenon – often modeled as random fields defined on a continuous domain – from a discrete set of data points. Kriging, based on Gaussian processes, is one of the commonly used tool for such inference problems. While Gaussian joint probability distributions have known closed form solutions, several physical phenomenon exhibit non-Gaussian features which are analytically intractable. In such problems, one often approximates the underlying distribution by some known, often simpler distribution (for example, a Gaussian), and infers an assigned parametric form for its moments. More rigorous analysis involves computationally expensive methods such as Markov Chain Monte Carlo (MCMC) methods. This paper presents a review of the diagrammatic perturbation theory (following Feynman diagrams used in Physics), a particular technique developed as part of Information Field Theory, for analytically estimating moments of perturbative non-Gaussian distributions.