Leveraging Feynman Diagrams for Statistical Inference in Engineering and Natural Hazard Mitigation

Bayesian Inference is becoming an increasingly popular method for various data-driven applications in engineering 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) simulations which are often highly expensive computationally. I will discuss the use of a "Diagrammatic Perturbation Theory" inspired from Statistical Physics and Information Theory, which allows us represent posterior moments 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 moments to a desired level of accuracy, it significantly outperforms MCMC methods in terms of speed. I will adapt and apply this method to tackle both parametric and non-parametric Bayesian inference on random fields and showcase our findings through a model that predicts hurricane-induced storm surge in the Gulf coast region.