Kernel Bayesian Inference with Posterior Regularization
We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of kernel Bayesian inference. Moreover, the optimization problem induces a new regularization for the posterior embedding estimator, which is faster and has comparable performance to the squared regularization in kernel Bayes' rule. This regularization coincides with a former thresholding approach used in kernel POMDPs whose consistency remains to be established. Our theoretical work solves this open problem and provides consistency analysis in regression settings. Based on our optimizational formulation, we propose a flexible Bayesian posterior regularization framework which for the first time enables us to put regularization at the distribution level. We apply this method to nonparametric state-space filtering tasks with extremely nonlinear dynamics and show performance gains over all other baselines.
This paper proposes theoretical results characterizing posterior mean embeddings as the solution of a vector-valued regression problem. As in RegBayes which formulates a posterior distribution (via Bayes' rule) as the solution to an optimization problem, this paper proposes kRegBayes which does the same to kernel Bayes' rule of Fukumizu et al., 2013. For computing a conditional mean embedding operator, a new regularization technique is also proposed and is shown to be faster and consistent. 
# Justifications of the ratings  Considering the complexity of this work which builds upon a number of existing technical papers, the authors did an excellent job in the writing. The writing is very clear and precise. The paper is self-contained. A potential academic impact from this work is from the optimizational formulation of kernel Bayesian inference which increases understanding and allows one to also add a regularization to the inferred posterior embedding. This can open up a new research direction for the kernel Bayes' rule as well as other techniques related to conditional mean embedding.  # Questions  1. Section 3.1 mainly deals with how to estimate the loss in (3) without using an i.i.d. joint sample to approximate the expectation (as we do not have it). What is mysterious to me is the absorption of \mu into f. Originally in (3), \mu is what we seek. Now in (4), the loss in (3) is rewritten in such a way that  the original \mu (a conditional mean embedding operator) is now in f, and a new \mu_{(X,Y)} appears (a joint embedding). I guess in the end f is treated as if it does not have \mu. Could you please elaborate a bit on what is going on here?   2. Line 152: f is assumed to be in a product RKHS. Is it really in the space? f looks like an unbounded function. What further conditions (e.g., on H_X and H_Y, or the domains) do we need for this to be true?  3. In theorem 3, does |\mathcal{X} \times \mathcal{Y}| < \infty mean the domains \mathcal{X}, \mathcal{Y} are discrete and not just that \mathcal{X} and \mathcal{Y} are bounded? If \mathcal{X}, \mathcal{Y} are indeed discrete, isn't this result fairly restrictive? Related: What are \mathcal{X}, \mathcal{Y} in the experiments?  4. Section 4 discusses a possibility of adding a regularization to the posterior embedding. I am not sure why the regularization (second term) in (10) is a sensible one. The description states that this is to encourage the posterior to be concentrated at some specified points. Why and when do we want this? I understand that this is just one way to do regularization under this framework. If there are more, what are others?   # Minor comments  * The proofs in the appendix are quite technical. It would be much better if   the paper contains a short proof sketch for each new result.  * Reference [14] has a wrong order of coauthors? * In section 2.2 and others, if I understand correctly \mu_{(YX)} is the same   as C^{\pi}_{YX}. It might make it a bit more readable to just use only C^{\pi}_{YX}. * Line 136: arg sup should be arg inf. * Line 234: By (30), you mean (10)?  ------------ after the rebuttal ---------- I read the authors' response.