Venue: CLeaR 2022 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: Summary: An interesting first step to model social influence
Main review: The authors describe PIF, a method to disentangle social influence from other underlying factors of prople's behavior. The question addressed is whether, knowing the connections between people, the fact that a user j bought an item yesterday makes user i buy an item today.

The authors posit a causal model which includes the social graph of "friend" connections, as well as confounders -- the features of the items, the individual preferences of users, the attributes that are responsible for their connection in the network and could also be responsible for their purchasing behavior. The authors propose a Poisson factorization model where the latent factors that are responsible for the graph structure -- independently of the purchases -- and the factors responsible for the previous day purchase. The authors propose an experimental validation with relevant baselines

The paper is well-written and overall clear. I liked the experiment section, and the idea that 


* temporal aspect: it seems that the model is correct under the assumption that yesterday's purchases are free of social influence. It seems unlikely to hold in practice. Is it possible to extend the model to account for multi-step dynamics that would be more relevant to the real world?
* assumption that there is no social influence if we do not observe a link: I don't see this assumption explicitly, except hidden in Eq 11, but it seems to be used implicitly (for instance in the choice of the definition of psi_{ijk}). Could the authors discuss this assumption and its relevance (e.g., on platform such as lastfm where the social component is unlikely to be very present). Also, why define psi_{ijk} conditionned on a_ij=1? It seems that removing this conditioning would be a valid definition of social influence as well. 
* assumptions for Proposition 1: it seems that some assumptions are missing. For instance, I suspect the result assumes that the true distribution follows the poisson model described in Eq 11 and that we have an infinite amount of data to estimate beta?
* "real-world" study: 
- admittedly the study is not really "real-world", in particular given the two-step dynamics. While I found it interesting that PIF obtains better performances (maybe) because other models overestimate the effect of social influence, maybe it feels like the approach only works in this case (because the assumption that yesterday's purchases do not depend on social influence approximately holds). 
- I don't have any order of magnitude in mind, but -127 for log-likelihood of counts of music pieces seems very low. What is the LL of a model that predics uniformly across all songs with the correct number of songs listened to as total average? 
* in applications where there is a recommender system which may use previous-day purchase history of "friends" to make recommendation (and thus influence today's purchases), to what extent the "social influence" we are trying to model is actually reverse-engineering the recommender system? Can the model be modified to take disentangle the influence due to the recommender system from a true "social influence" that would be due to social interactions between friends?

other remarks:
Figure 1:
* the legend uses x_{ik} but the figure x_{jk}
* the edge x_{jk} -> rho_j is in the wrong direction
* the edge x_{jk} -> y_{ik} is missing
review_title: null

final_score: rating: 5: Marginally below acceptance threshold
review_content: Summary: This work extends Deconfounder with a Poisson factorization model to estimate social influence in network observational data.
Main review: This work aims to estimate social influence, a typical type of spillover effect in social networks. It proposes PIF (Poisson Influence Factorization), which takes social network, past and current behaviors of members as its input. It works in a similar way to Deconfoudners, which first infers latent substitutes of hidden confounders and then do a linear regression to estimate potential outcomes. Semi synthetic experiments show the proposed method PIF-Net and PIF-Joint outperforms existing methods and can achieve similar performance to Oracle. The real-world experiment further verifies PIF-joint's effectiveness through a prediction task. 

Major concern:
1. It is not quite clear why it is needed to condition on a_i^{-j} and x_i^{-j} given the fact the only confounder for the causal effect of x_jk on y_ik in fig. 1 is \tau_k.
2. It seems there is no strong motivation behind the poisson factorization model, can we replace it with other types of matrix factorization or neural networks?
3. In the real-world experiment, it is not very fair to use Poisson likelihood as the performance metric since we do not know whether the underlying data generating process follows the Poisson factorization model. I would suggest to directly show the accuracy or MSE of predicted outcomes or predicted treatments.

Comments:

1. The causal graph in Fig. 1 cannot show x_jk is prior to y_ik in time.
2. Similarly, in Eq.(1), I believe we also need to emphasize the value of y_ik must be determined after do(x_jk).
3. It is better to make it clear what is the expectation in eq.(3) taken over.
4. For eq.(8) to hold, it is better to specify that it needs to assume v_i and u_i are exogenous.
5. It is better to explain the difference between c_i and z_i, it seems both of them only influences edges.
6. In eq.(9), why x_ik is not affected by any social influence?
7. What is the difference between eq.(10) and the second term of eq.(9)? Both of them factorize x to latent variables.
review_title: null

final_score: rating: 6: Marginally above acceptance threshold
review_content: Summary: The paper introduces a causal model for social influence in order to estimate social influence when there are unobserved confounders. The idea is that there is an observed  social network that determines the relationship between individual "buyers", and unobserved attributes that affect both the connections between buyers and items that are bought. The model also assumes the there exist only two different times (called yesterday and today), which attribute to the causal interpretation of the model. In order to estimate the social influence, a joint factor Poisson model is fitted for certain unifying substitutes of the confounders, and then these substitutes are estimated. There is a simulation study that shows under the setting of the simulation, this model (when fitting a joint model of purchases and the network data) performs better than two other alternative models. There is also a real world data analysis of the model using the data collected from a song-sharing platform.
Main review: The paper is clearly written, and the technical details seem correct to me. I appreciate the use of Poisson model for social influence -- although the model itself as well as the use of a single latent variable instead of several of them in graphical models are not novel, the idea of a joint model for both network connections and purchases is quite interesting. However, I have some concerns and confusions about the model as I describe below:

Firstly, the idea of two time points (today and yesterday) is rather strange to me. Of course the notion of time here (perhaps too conveniently) imposes a causal direction between two nodes of the network. However, it completely disregards the fact that, when correlation is observed and the effect of confounders are no longer a concern (as it is the case here), the causal direction could be in either direction: what if the day after the allegedly cause node buys the item again -- can not the effect be considered a cause with exactly the same setting?

In addition, the idea of hypothetical intervention is rather confusing to me. It seems to me in reality a counterfactual distribution is being computed not an interventional one, but this should be made clear regardless. More importantly, the use of backdoor criterion to relate the observational and interventional distributions is questionable here as firstly the causal graph is just assumed (not learned) and secondly the Markov property is not formally shown. 

The causal graph of Figure 1 looks rather strange to me as well. Since $a_{ij}$ is a collider node in the graph, the model is basically saying that, given the relationship between $i$ and $j$ in the network (regardless of whether they are connected or not), The nodal attributes are correlated,and consequently, after them being marginalized over, $x_{jk}$ and $y_{ik}$ have some symmetric association (which is usually indicated by an undiredcted edge in mixed graphical modeling). I do not think this is what the authors intend for their model. At some stage (when discussing Figure 2), it seems that the assumption, when nodes are at the same time point, is dyadic independence given the attributes, but the situation is more complicated as there are two different time points in the model. Moreover, the graphical model is only described for two nodes in the network. How does this expand when one considers the whole network, and how does the social influence is measured for nodes that are not adjacent by connected via a path in the graphical models? In general, the independence structure of the whole model is not clear to me.

There are also many assumptions especially when using the Bayesian estimation for substitutes. There should be a justification of these assumptions, or at least a discursion on them.

Finally, my problem with the simulation study is that, by design, the purchases are happening where the connections in networks are. This is rather counterintuitive when one wants to test a joint model for purchases and network connections. I think there should be a variety of scenarios regarding purchases to make this simulation study more reliable.
review_title: null

Meta-review: This is an interesting contribution to an important problem in causal inference.  The problem setting has a number of unusual features, as noted by reviewers, most notably limiting itself to two time periods, with no confounding in the first period, but these are stylized features of the literature which everyone agrees are unrealistic, and hope will be removed once we as a field have a better handle on the simplified setting. (We'll see!)

Within that context, the manuscript makes a real advance over prior work (e.g. McFowland & Shalizi) by using more flexible parametric modeling assumptions, which allow for simultaneously modeling multiple related behaviors diffusing over the network (while e.g. S&McF. only consider a single, scalar-valued behavior).  The manuscript also clarifies the definition of the to-be-identified causal estimand of social influence, rather than just a model parameter.  The proposed method does well in a simulation example (using a realistic network) where the parametric model is exactly correct (sensitivity to, e.g., non-Poisson distributions weren't considered).  The behavior on a real-data example is promising, but the method's assumptions are plainly violated there (*), and, as usual, there's no ground truth about influence here, only held-out predictions.

One final limitation to the manuscript is that methods like this can only deliver in-the-limit identification (with finite samples, there is always some bias, because estimated substitutes are never quite as informative as true latents), and it is not quite clear to the reader what the right limit is here.  With the M&S paper, the correct limit is the number of people $n\rightarrow\infty$, with the number of time periods $T$ being either finite or growing.  In the present manuscript, I think we need both the number of people $n\rightarrow\infty$ and the number of items/behaviors $m\rightarrow\infty$, though there is no formal consistency/convergence analysis so this is unclear.

Overall, I believe this is a correct, original, clear and useful contribution to the problem of inferring social influence from observational network data.  It is an advance, and it presents fruitful directions for further work.

*: because the real data extends over time, "yesterday" is confounded by "day before yesterday", etc.; the sub-sampled graph means that there are lots of ignored social pathways between retained nodes via deleted nodes; selecting nodes by degree biases all the retained nodes to have similar latent attributes (viz., those implied by high degree), which can make homophily appear to be less important; etc.  Since the referees did not address this I will not belabor these points but I feel it important to bring them to the authors' attentions.
Decision: Accept (Poster)