Kernel Bayes rule

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.

💡 Research Summary

The paper introduces the Kernel Bayes Rule (KBR), a non‑parametric framework that realizes Bayes’ theorem entirely within reproducing kernel Hilbert spaces (RKHS). The key idea is to represent any probability distribution by its mean embedding μ_P = E_{X∼P}