Cluster Variational Approximations for Structure Learning of Continuous-Time Bayesian Networks from Incomplete Data
Continuous-time Bayesian networks (CTBNs) constitute a general and powerful framework for modeling continuous-time stochastic processes on networks. This makes them particularly attractive for learning the directed structures among interacting entities. However, if the available data is incomplete, one needs to simulate the prohibitively complex CTBN dynamics. Existing approximation techniques, such as sampling and low-order variational methods, either scale unfavorably in system size, or are unsatisfactory in terms of accuracy. Inspired by recent advances in statistical physics, we present a new approximation scheme based on cluster variational methods that significantly improves upon existing variational approximations. We can analytically marginalize the parameters of the approximate CTBN, as these are of secondary importance for structure learning. This recovers a scalable scheme for direct structure learning from incomplete and noisy time-series data. Our approach outperforms existing methods in terms of scalability.
The paper introduces a generalization of previous variational methods for inference with jumps processes; here, the proposal approximating measure to the posterior relies on a star approximation. In application to continuous-time Bayesian networks, this means isolating clusters of nodes across children and parents, in order to build an efficient approximation to the traditional variational lower bound. The paper further presents examples and experiments that show how the proposed approach can be adapted to structure learning tasks in continuous-time settings.  This is an interesting and topical contribution likely to appeal to the statistical and probabilistic community within NIPS. The paper is, in overall, well-written and reasonably well-structured. It offers a good background on previous work, helps the reader to understand its relevance and put its results in context within the existing literature. For these reasons, I wish to vote for weak acceptance. However, I have two major comments regarding rigor and content within some parts of the text. Specifically,  1 - The very first formulae presented around equation (1) seems wrong, unless I am missing something trivial. I pressume the negative lnP(Y) on the above equation has to be added, not substracted. As these are fairly common concepts and inequalities, this mistake sheds a lot of doubt to the reader, when approaching the sheer amount of calculations and formulae typical in such variational works.  2 - Linked to the above and in terms of reproducibility of the results. It is good to see in Section 3 that differential equations for the entropy and energy simplify to previously presented results in the literature. However, the bit that justifies acceptance in my opinion is the structure learning component in Section 4. In this case, pretty much all derivations are left to the appendices, the thin is rushed and feels hard to parse. The average reader will likely never look at appendices and this reduces accessibility. While it is understandable that not everything can fit, it would be valuable to add further analytic intuition behind formulae in (7)-(8).  Finally, there exist minor typos that warrant a re-read.  -----------  After rebuttal:  I appreciate the clarifications from the authors and still think this is an above average contribution likely to spark interest. In my opinion 6 is still an appropriate score.