## Kernel Bayes' Rule: Bayesian Inference with Positive Definite Kernels

*Kenji Fukumizu, Le Song, Arthur Gretton*; 14(Dec):3753−3783, 2013.

### Abstract

A kernel 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 Bayesian computation without likelihood and
filtering with a nonparametric state-space model.

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