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Kernel Bayes' Rule: Bayesian inference with positive definite kernels


Kenji Fukumizu

The Institute of Statistical Mathematics Department of Statistical Modeling



Kernel Bayes' Rule: Bayesian inference with positive definite kernels.


I will present a general method for Bayesian inference with positive definite kernels.  First, I will explain how positive definite kernels can be used for discussing conditional probabilities with covariance operators on reproducing kernel Hilbert spaces (RKHS). This is based on a recent development which uses kernels for dealing higher order statistics by embedding distributions in the form of means in RKHS.

Second, it will be shown that the Bayes' rule is realized as operations on the kernel mean expression of the conditional probability and the prior.

The kernel mean of the posterior is then obtained by Gram matrix computation to realize the procedure of Bayes' rule. The rate of convergence of the empirical kernel estimate to the true posterior is also derived. Finally, as applications, I will show some experimental results on kernel nonparametric HMM and Bayesian computation without likelihood.