Efficient Posterior Estimation for Perturbative Non-Gaussian Distributions Using Feynman Diagrams

Bayesian inference is a popular method for various data-driven applications in a variety of fields due to its ability to incorporate prior knowledge and uncertainty directly into the model. However, obtaining closed-form solutions for posterior distributions and its moments is challenging, often requiring either analytical approximations that may not capture the underlying distribution with fidelity, or numerical methods like Markov Chain Monte Carlo (MCMC) which are often highly expensive computationally. We discuss the use of a "Diagrammatic Perturbation Theory" inspired from Statistical Physics and Information Theory, which allows the computation of the posterior of a perturbative non-Gaussian distribution as a sum over a set of diagrams (also known as Feynman diagrams), where each diagram is a shorthand for a mathematical expression constructed out of analytically known, simpler quantities. In addition to giving closed-form expressions for posterior and its moments to a desired level of accuracy, it significantly outperforms MCMC methods in terms of speed. We adapt and apply this method to tackle parametric Bayesian inference on random field models such as those used for predicting hurricane-induced storm surge.