Kernel Bayes' rule

TL;DR

This paper introduces a nonparametric kernel-based approach to Bayesian inference using reproducing kernel Hilbert spaces, enabling Bayesian computations without explicit parametric models.

Contribution

It proposes a novel kernel Bayes' rule framework that represents probabilities as RKHS means, allowing Bayesian updates directly from empirical samples without parametric assumptions.

Findings

Consistent estimator for posterior expectations derived

Applications include Bayesian computation without likelihood

Effective filtering with nonparametric state-space models

Abstract

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on representations of probabilities in reproducing kernel Hilbert spaces. Probabilities are uniquely characterized by the mean of the canonical map to the RKHS. The prior and conditional probabilities are expressed in terms of RKHS functions of an empirical sample: no explicit parametric model is needed for these quantities. The posterior is likewise an RKHS mean of a weighted sample. The estimator for the expectation of a function of the posterior is derived, and rates of consistency are shown. Some representative applications of the kernel Bayes' rule are presented, including Baysian computation without likelihood and filtering with a nonparametric state-space model.