A geometric ensemble method for Bayesian inference

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.