🤖 AI Summary
This work addresses the limitations of conventional Gaussian approximations and particle filters in Bayesian inference for non-Gaussian, multimodal posterior distributions. We propose a geometric Bayesian inference framework based on ensembles of uniform distributions over convex polytopes. Our method models the posterior as a recursively updated collection of convex polytopes, each supporting a uniform distribution—marking the first such formulation. Crucially, we introduce a Kalman-style polytope update mechanism that integrates ensemble filtering principles with convex geometric operations, including Ikeda mappings and Minkowski sums. Unlike moment-based or point-estimate approaches, our method inherently captures support-set structure and uncertainty propagation without relying on statistical moments. Experiments on the low-dimensional Ikeda system and high-dimensional Lorenz’96 system demonstrate substantial improvements in modeling accuracy and numerical stability for non-Gaussian posteriors, thereby extending the expressive capacity beyond classical approximation methods.
📝 Abstract
Conventional approximations to Bayesian inference rely on either approximations by statistics such as mean and covariance or by point particles. Recent advances such as the ensemble Gaussian mixture filter have generalized these notions to sums of parameterized distributions. This work presents a new methodology for approximating Bayesian inference by sums of uniform distributions on convex polytopes. The methodology presented herein is developed from the simplest convex polytope filter that takes advantage of uniform prior and measurement uncertainty, to an operationally viable ensemble filter with Kalmanized approximations to updating convex polytopes. Numerical results on the Ikeda map show the viability of this methodology in the low-dimensional setting, and numerical results on the Lorenz '96 equations similarly show viability in the high-dimensional setting.