(Section 11). Lemma 2. Suppose the prior satisfies Lemma 3. Suppose the prior satisfies Lemma 4. Suppose the prior satisfies This is the most difficult technical step in establishing our result. Lemma 5. Consider the event We briefly explain the heuristic idea behind the proof of Lemma 5. Since the VB posterior We also work with the default parametrization of the bmlasso function of the BhGLM package, i.e. We provide five further test cases in addition to the experiment considered in Section 5. In all cases we consider Gaussian design matrices, but vary all other parameter. We ran each experiment 200 times and report the means and standard deviations of the performance metrics in Table 3.

Since the VB algorithm does not depend on the unknown truth to achieve optimality, our results shed light on effective prior choices.

We thank the reviewers for their constructive suggestions. A summary of the added discussion is provided point-by-point below. Reviewer 1: How much more work is needed versus linear regression. The technical details are thus different (and more involved) here. We have included these derivations for completeness.

It is not a surprise that kernel methods are among the most theoretically studied models. From a numerical point of view, they reduce to convex optimization problems that can be solved with strong guarantees.

The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals.

By combining robust regression and prior information, we develop an effective robust regression method that can resist adaptive adversarial attacks. Due to the widespread existence of noise and data corruption, it is necessary to recover the true regression parameters when a certain proportion of the response variables have been corrupted. Methods to overcome this problem often involve robust least-squares regression. However, few methods achieve good performance when dealing with severe adaptive adversarial attacks. Based on the combination of prior information and robust regression via hard thresholding from [ 1 ], this paper proposes an algorithm that improves the breakdown point when facing adaptive adversarial attacks. Furthermore, to improve the robustness and reduce the estimation error caused by the inclusion of a prior, the idea of Bayesian reweighting is used to construct a more robust algorithm. We prove the theoretical convergence of proposed algorithms under mild conditions. Extensive experiments show that, under different dataset attacks, our algorithms achieve state-of-the-art results compared with other benchmark algorithms, demonstrating the robustness of the proposed approach.