A geometric ensemble method for Bayesian inference

TL;DR

This work develops a geometric framework for Bayesian inference using uniform distributions on $\mathcal{H}$-polytopes, introducing a family of convergent ensemble filters (CPF, ECPF, KCPF, EKCPF) and their ensemble variants (BCPF, EnCPF, EnKCPF). It combines polytope-based priors/posteriors with Kalman-inspired updates, kernel-density mixtures on $\mathcal{H}$-polytopes via $\omega$-points, and hit-and-run resampling to form scalable, convergent filters for data assimilation. The approach is validated through sequential experiments on the Ikeda map and Lorenz '96, showing competitive performance across low and high dimensions and illustrating the trade-offs between conservative polytope approximations and Kalmanized updates. The paper highlights practical scalability and outlines future directions, including improved priors for the ensemble, localization, and incorporation of constraints to enhance performance in high-dimensional Bayesian inference tasks.

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.

Figures (5)

• Figure 1: Evaluation of the CPF (left panel) and the ECPF (right panel) on their respective test problems. On both panels, the left blue dotted rectangle represents the prior uniform distribution, the red dashed polyhedron (left panel) and doughnut (right panel) represent their respective measurement distributions, the solid yellow outline represents the exact posterior distribution, and the dash-dotted polytope (right panel) represents the ECPF approximation.
• Figure 1: Evaluation of the KCPF (left panel) and the EKCPF (right panel) on their respective test problems. On both panels, the left blue dotted rectangle represents the prior uniform distribution, the red dashed outlines represent their respective measurement distributions with each line representing one, two, and three standard deviations, the solid yellow outline represents the exact posterior distribution again with one, two, and three standard deviations, and the dash-dotted polytopes represents the KCPF and EKCPF approximations in the left and right panels respectively.
• Figure 1: Evaluation of the EnGMF (top left panel), the BCPF (top right panel), the EnCPF (bottom left panel), and the EnKCPF (bottom right panel) on the static test problems. On all panels, the left blue dotted ovals represent one, two and three standard deviations of prior normal distribution, the solid blue dots represent the prior samples, and the solid blue outlines represent the kernel density estimates using the samples. The red dashed outlines represent the measurement distribution with each circle slice representing one, two, and three standard deviations. The solid yellow outline resembling a banana represents the exact posterior distribution again with one, two, and three standard deviations. The dash-dotted purple outlines represent the respective posterior approximations by their respective algorithms.
• Figure 1: Ensemble size ($N$) versus mean spatio-temporal RMSE for the Ikeda map with the EnGMF (blue line with circular markers), EnKF (read line with square markers), BCPF (cyan line with plus markers), BPF (green line with x markers), and the EnKCPF (yellow line with star markers). Additionally, a 'no filter' scenario is plotted as the top dashed line, and a true Bayesian inference scenario is plotted as the bottom dotted line.
• Figure 2: Ensemble size ($N$) versus mean spatio-temporal RMSE for the Lorenz '96 equations with the EnGMF (blue line with circular markers), EnKF (red line with square markers), and the EnKCPF (yellow line with star markers).

Theorems & Definitions (19)

• Remark 2.1: $\mathcal{H}$-polytope representation of the Minkowski sum
• Remark 2.2
• Remark 3.1: Propagation in the CPF
• Lemma 3.2: Measurement inverse approximation
• Proof 1
• Theorem 3.3: Extended Measurement Polyhedron Approximation
• Theorem 4.1: Kalmanized convex polytope filter
• Proof 2
• Corollary 4.2
• Theorem 5.1