Reviews: Kernel Bayesian Inference with Posterior Regularization

This paper provides an interesting connection between kernel Bayesian inference and vector valued regression. Based on this, a new regularization method is provided to compute an approximation of the kernel embedding of the posterior distribution. Simulation results look promising, suggesting that the new method gains improvement over many existing methods. However, as a non expert, from reading the current introduction, I'm still confused about the motivation of using kernel Bayesian inference---in order to approximate the kernel embedding of the posterior, a sample of iid draws (x_i, y_i) from the joint distribution of the parameter/hidden variable (X in the paper) and data (Y in the paper) are assumed to be available. First, it is a highly non-trivial problem of obtaining samples (x_i)'s from the posterior.