Geometric Variational Inference and Its Application to Bayesian Imaging
Philipp G. Frank · 2022
Modern day Bayesian imaging problems in astrophysics as well as other scientific areas often result in non-Gaussian and very high-dimensional posterior probability distributions as their formal solution. Efficiently accessing the information contained in such distributions remains a core challenge in modern statistics as, on the one hand, point estimates such as Maximum a Posteriori (MAP) estimates are insufficient due to the nonlinear structure of these problems, while on the other hand, posterior sampling methods such as Markov Chain Monte Carlo (MCMC) techniques may become computationally prohibitively expensive in such high-dimensional settings. To nevertheless enable (approximate) inference in these cases, geometric Variational Inference (geoVI) has recently been introduced as an accurate Variational Inference (VI) technique for nonlinear unimodal probability distributions. It utilizes the Fisher–Rao information metric (FIM) related to the posterior probability distribution and the Riemannian manifold associated with the FIM to construct a set of normal coordinates in which the posterior metric is approximately the Euclidean metric. Transforming the posterior distribution into these coordinates results in a distribution that takes a particularly simple form, which ultimately allows for an accurate approximation with a normal distribution. A computationally efficient approximation of the associated coordinate transformation has been provided by geoVI, which now enables its application to real-world astrophysical imaging problems in millions of dimensions.