Learning Large-Scale Poisson DAG Models based on OverDispersion Scoring
In this paper, we address the question of identifiability and learning algorithms for large-scale Poisson Directed Acyclic Graphical (DAG) models. We define general Poisson DAG models as models where each node is a Poisson random variable with rate parameter depending on the values of the parents in the underlying DAG. First, we prove that Poisson DAG models are identifiable from observational data, and present a polynomial-time algorithm that learns the Poisson DAG model under suitable regularity conditions. The main idea behind our algorithm is based on overdispersion, in that variables that are conditionally Poisson are overdispersed relative to variables that are marginally Poisson. Our algorithms exploits overdispersion along with methods for learning sparse Poisson undirected graphical models for faster computation. We provide both theoretical guarantees and simulation results for both small and large-scale DAGs.
The paper considers the inference of the structure of the DAG underlying a Poisson directed graphical model. The proposed strategy relies on the fact that nodes with parents are conditionally Poisson, whereas nodes without parents are marginally Poisson. The paper aims at distinguishing the former from the later based on an over-dispersion criterion.    I do like the principle of the approach but the construction is only partly convincing. My main concerns are the following:   1 - I am not completely clear about the proof if identifiability. The proof of Thm 3.1. starts with "assume that the true causal ordering \pi^* is {1, 2, ...p}". To me, in a general oriented graphical model, there may be several true causal orderings. As the proof relies on induction, this should be clarified.   2 - The exploration of the DAG space is indeed a critical task and the authors simply suggest to use GLMlasso at each step to reduce the search space. The objective function of the GLMlasso is not necessarily consistent with the overdispersion criterion, so the global procedure is a bit heuristic.   3 - In the simulation study, I did not understand why all regression coefficients $\theta_{jk}$ are chosen to be negative (between -1 and -.7). This is allthemore surprising as the authors states that Yang requires this negativity constraint, while they do not. Does this choice make the problem easier?   4 - As for the performances, indeed they are good will large $n$, even for fairly large $p$, by I have been very surprised by the bad performances for $p = 10$ and $n = 500$ which seems to be a rather comfortable case. Do the authors have an idea why they fail in such a, apparently, easy situation?        Minor remarks:   1 - In the proof of Thm 3.1 should "Pa(j)/{1, 2, ..., m}" be "Pa(j)\{1, 2, ..., m}"?   2 - The threshold $c_0$ is likely to have some influence on the results, which is not explored in the simulation study.