Kernel Bayesian Inference with Posterior Regularization
This work provides a novel regularization framework for kernel Bayesian inference, enabling distribution-level regularization for the first time, which is significant for researchers in Bayesian nonparametrics and kernel methods.
The authors tackled the problem of kernel Bayesian inference by proposing a vector-valued regression formulation equivalent to RKHS embedding of Bayesian posteriors, which provides a new understanding and enables distribution-level regularization. They demonstrated performance gains in nonparametric state-space filtering with extremely nonlinear dynamics, outperforming all other baselines.
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.