A nonparametric kernel-based method for realizing Bayes ’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes ’ rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric statespace model. A consistency rate for the posterior estimate is established. 1

Constructing tractable dependent probability distributions over structured continuous random vectors is a central problem in statistics and machine learning. It has proven difficult to find general constructions for models in which efficient exact inference is possible, outside of the classical cases of models with restricted graph structure (chain, tree, etc.) and linear-Gaussian or discrete potentials. In this work we identify a graphical model class in which exact inference can be performed efficiently, owing to a certain “low-rank ” structure in the potentials. While we focus on the case of tree graphical models, the lowrank treatment can also be applied for efficient exact inference in certain sparsely-loopy models. We explore this new class of models by applying the resulting inference methods to neural spike rate estimation and motioncapture joint-angle smoothing tasks. 1