Remarks on kernel Bayes’ rule

The kernel Bayes’ rule has been proposed as a nonparametric kernel-based method to realize Bayesian inference in reproducing kernel Hilbert spaces. However, we demonstrate both theoretically and experimentally that the way of incorporating the prior in the kernel Bayes’ rule is unnatural. In particular, we show that under some reasonable conditions, the posterior in the kernel Bayes’ rule is completely unaffected by the prior, which seems to be irrelevant in the context of Bayesian inference. We consider that this phenomenon is in part due to the fact that the assumptions in the kernel Bayes’ rule do not hold in general.