Kernel Bayesian Inference with Posterior Regularization

TL;DR

This paper introduces a new kernel Bayesian inference method using posterior regularization, providing faster algorithms with theoretical guarantees and improved performance in nonlinear state-space filtering tasks.

Contribution

It presents a novel regularization framework for kernel Bayesian inference that operates at the distribution level, with theoretical consistency analysis and practical benefits.

Findings

Faster regularization method with comparable performance to existing approaches

Theoretical proof of consistency for the proposed regularization

Improved filtering performance in nonlinear state-space models

Abstract

We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of kernel Bayesian inference. Moreover, the optimization problem induces a new regularization for the posterior embedding estimator, which is faster and has comparable performance to the squared regularization in kernel Bayes' rule. This regularization coincides with a former thresholding approach used in kernel POMDPs whose consistency remains to be established. Our theoretical work solves this open problem and provides consistency analysis in regression settings. Based on our optimizational formulation, we propose a flexible Bayesian posterior regularization framework which for the first time enables us to put regularization at the distribution level. We apply this method to nonparametric state-space filtering tasks with extremely nonlinear dynamics and show performance gains over all other baselines.