The authors present a variational EM algorithm for inference in a multivariate Hawkes process. This allows learning in the setting with many hyper-parameters and limited data, where ML approaches will fail or at least hyper-parameter estimation would be difficult. ML approaches therefore tend to use less hyper-parameters which may limit their performance. This class of model has seen a lot of interest recently and I’m not aware of a previous approach to variational inference in the continuous-time setting. A previous stochastic variational inference approach from Linderman and Adams (cited here) used a discrete-time approximation and other Bayesian approaches that I’m aware of are based on MCMC. This therefore looks like a timely and valuable contribution. Experiments are run to look at performance in terms of link and event prediction and for both parametric and non-parametric excitation function models. The VI approach is shown to outperform ML approaches on limited data and looks better even for pretty large datasets according to the synthetic data evaluation. VI also seems to deal better with an over-complex non-parametric function compared to ML. Results on three real datasets also look very good compared to MLE approaches.

Post rebuttal: Many thanks for addressing my concerns. I have read the rebuttal and other reviewer's comments. Please elaborate them in more details as needed in the revision. 1. Regarding (8) -> (9) transformation: perhaps there could be a data-dependent way of learning a better prior of \alpha. 2. Regarding my comment that the case where one dimension has scarce data. The scarcity of the data is caused by the naturally smaller intensity function. If one dimension has a harder to learn triggering function, it may propagate onto other dimensions. In prior works, such cases are hard to handle, and I think the method proposed here could be used as a remedy to this issue. I don't think the intuition explanation is a satisfying answer and I believe some additional exploration is warranted here. Overall I maintain my vote to accept the paper. -------------------------------------------------------------------------------------- This paper proposes learning the parametric MHP by regularized MLE. The parameters to be learned are \mu and W, and a limitation of the traditional algorithms are that all the dimensions share the same regularization parameter. The authors proposed a Bayesian method and modeled \alpha, the regularization coefficient, as a hidden variable. An EM algorithm follows naturally. Experimental results show that the proposed method improves upon the EM-based learning algorithm MLE-SGLP proposed by Xu et al. I feel like this is a good submission but it's quite possible I didn't understand the main part of the paper. I think the idea is novel and empirical evidence suggests it would help researchers to select the regularization parameters. In cases where learning samples is scarce, I think this is particularly useful. Empirical evidence suggests that the method works, but formulation-wise, there are a few places that confused me although I already think it's quite interesting. 1. The first key step in the paper is from (4) to (8), where the regularizer is viewed as \log p_{\alpha}(\mu,W). I would argue that the ignored normalization coefficient here depends on \mu and W itself, and cannot be omitted. Or \alpha has to depend on \mu and W (which is not the case per Appendix). Therefore I suppose this is not an equivalent transformation? I also think it is this reason that leads to the fact that one cannot optimize \alpha in (4) directly, am I right for this? I didn't understand why one can optimize over (8) (stated in L142) as one can change \alpha arbitrarily and it doesn't affect the choice of the parameters which is determined by the problem. 2. I didn't quite understand the intuition behind the transformation from (8) to (9). In L147 I suppose the upper bound should be the maximum likelihood instead of the likelihood? 3. I saw in appendix several choices of p_{\alpha}. How did you obtain this from (4)? Any reasons why p_{\alpha} should be designed as described in Appendix? Understanding those along with other reviewer's concerns would help me better appreciate the result of this paper.

In the inference problem of multivariate Hawkes processes, the regularization is critical for avoiding over-fitting when only short sequences of events are available. This paper develops the variational EM algorithm, which allows for efficiently determining a lot of hyper-parameters that control the influence of regularization penalty. Also, it can estimate the model parameters while considering the uncertainty of them. This paper is well-written and easily understandable. The proposed algorithm is practically useful but seems to be a straightforward application of the existing techniques. Comments following the guidelines as requested. [Originality] Strengths. The authors make use of the black-box variational inference to effectively determine the hyper-parameters for controlling the regularization penalty. Weaknesses. My concern is that the proposed algorithm seems somewhat incremental, as it is a combination of well-known techniques. I think It would be interesting to consider the case of more high dimensional data, that is, the number of nodes is large. In that case, the authors might need to stretch the algorithm so that the computational complexity can be reduced. [Quality] Strengths. I think this paper is technically sound, and the experimental section is well-organized. There is convincing experimental evidence, i.e., analysis of the uncertainty, in Figure 3. Weaknesses. The limitations of the proposed approach are not discussed. How is the efficiency and execution time of inference? Could the authors provide enough experimental results or theoretical analysis? Especially, in all the experiments, the authors consider the case of relatively low dimensional data (up to 143). It would be a good idea to discuss the case of applying to more high dimensional data; is it applicable in terms of computation costs? The number of nodes might be larger in typical applications such as social-network studies (e.g., [1]). Several questions are as follows: 1) Why don't you optimize the decay-parameters? 2) Could the authors visualize the estimated causal-networks? I would like to see the networks that are estimated by the MLE-based method and the proposed method, respectively. [Clarity] This paper is easy to understand and provides enough information to implement. Small suggestion: In the re-parameterization of $\mu$ and $W$, the superscript notation (e.g., $\epsilon^\mu$) might be confusing. It would be better to denote them as alternative notations, e.g., $\epsilon_\mu$. [Significance] The inference problem of multivariate Hawkes processes under short observation sequences is an important task, and this paper has a useful contribution, easy to implement. However, I worry whether an application to the large-scale data sets whose number of nodes is large is difficult in practice. I think it would be significant to address the case of high dimensional data and provide analysis of the estimated causal-structures from the large-scale graphs. [1] T. Iwata et al., Discovering Latent Influence in Online Social Activities via Shared Cascade Poisson Processes, KDD, pages 266–274, 2013. ------------------------------ After author feedback: I appreciate the responses to my questions. The new experimental results in the rebuttal is a welcome addition. On the other hand, I think It would be a good idea to add the comparisons of the estimated excitation matrix $W$ and clarify the effect of the reguralization. Also, I still think that the variational inference scheme derived by the authors is a combination of several ML techniques; its technical contribution is somewhat low. But I think the proposed algorithm is effective and has a good impact for ML community and practitioners. Overall, my score for the paper remains the same.