NeurIPS 2020

A Robust Functional EM Algorithm for Incomplete Panel Count Data

Review 1

Summary and Contributions: The paper deals with the problem of missingness in panel count data. Specifically, an algorithm for pointwise estimation of the mean function of a Poisson point process in the case of censored counts is presented. The idea is to exploit the expectation-maximization scheme to overcome the problem of missingness: at every E step of the functional EM algorithm, the missing counts are substituted by the mean counts according to the mean function approximation from the previous M step. Data are assumed to be missing-completely-at-random. After the presentation of the algorithm, the work is devoted to provide necessary hypotheses to ensure convergence. In a variational framework (Gateaux derivatives), the contractiveness of the procedure is proved, with specific assumptions on variation of the target function and distance of the initialization. A rate of convergence in probability is given for the sample algorithm. Moreover, the paper includes examples of application to synthetic data, real data with synthetic missingness, and real censored data. In the first analysis, a deviation from the Poisson distribution for the generating counting process is explored. In the second a (slight) deviation from the complete randomness hypothesis is studied. The real data consist in ecological momentary assessment about the number of smoked cigarettes by smoke-quitters since the last request. In this context, long interval reports are considered unreliable by behavioral scientists. Indeed, an estimation of the mean function treating long intervals counts as missing (using the presented algorithm), highlights an (expected) under-reporting. I have read the authors' feedback, and appreciated their effort to elaborate further on the aspects highlighted in the reviews. I confirm my original mildly positive evaluation.

Strengths: The work extends the relatively limited state-of-the-art methodology to process panel count data in presence of missing counts. It provides theoretical guarantees for EM procedures when missing or unreliable records affect panel counts, which may be the case in many real datasets. The motivation for treating data coming from self-report via ecological momentary assessment is interesting and up to date. The work is a straightforward and needed contribution that facilitates the investigation of panel count data.

Weaknesses: The relevant contribution of the work is limited to finite-samples guarantees of the point estimation of the mean function. No uncertainty quantification can be conducted with the proposed method, which is a major drawback. The results are based on many simplifying assumptions. However, the main limits are recognized in the paper. The less convincing argument regards the performance of the procedure under deviations from the missing-completely-at-random assumption. Such deviations are investigated through a single example where the missing generation process slightly deviates from the missing-completely-at-random assumption, leaving many doubts on the performance of the procedure in more realistic scenarios. Moreover, it would have been useful if the authors had provided more insights, as well as a verification procedure, on the conditions that links c, the uniform lower bound on the increments of the true mean function, and r, the radius of the ball inside which the function initialization should lie.

Correctness: Statements and proofs are, overall, correctly carried out. A minor remark for first line of Proof of Lemma 1, in the supplementary material: after the first inequality, the summing index j should not be there anymore, is there a max missing?

Clarity: The paper is overall clearly written. Some minor remarks are the following. - The symbol $\tau$ is used both for endpoint of the time interval and for the missingness flag vector. - The notation $\Delta N^{(\tau)}$ for the elementwise product of $\Delta N$ and $\tau$, resulting in a vector of counts and zeros whether the count is missing, (or the complementary $\Delta N^{(s)}$) is never used again. - Assumption 7 in Section 5.1 is the only one missing an explanation (except in the supplementary material, where it is used in proofs) that may address its acceptability in applications.

Relation to Prior Work: The authors are giving full credit to what is already treated both in the panel count data literature and in the EM convergence one, also in matter of proving techniques.

Reproducibility: Yes

Additional Feedback: The work appears properly carried out and provides a few important results. However, the main idea of using the EM procedure on missing data is not original, the convergence results exploit many assumptions, and the empirical evaluations used to check deviations from the assumptions are limited. Interesting improvements would be: the introduction of some uncertainty quantification technique (to perform tests and compute confidence bounds), a deeper investigation of the performance under other and more realistic missingness mechanisms, and the introduction of some procedure to check whether the number of missing points in a certain dataset is too high to apply the proposed technique.

Review 2

Summary and Contributions: This submission introduces an EM algorithm to estimate the mean parameter of a Poisson process in the presence of missing values. The main motivation is incomplete panel count data, where users self-report a health-related behavior (such as smoking a cigarette after quitting) using an app. The submission provides asymptotic and finite sample convergence properties for the estimator and illustrate its behavior on two datasets.

Strengths: The submission is well written, very clearly presented and easy to follow. It provides a lot of intuition about the content of the paper, making it possible to grasp a first general idea without diving into the notation or technical results, and making it easier to understand these technical results in a second step. Dealing with incomplete panel count data seems like an important and unaddressed topic, which this submission is the first to solve.

Weaknesses: As mentioned by the authors, MCAR is a strong assumption. It is mentioned in the discussion and a theoretical assessment or alternative estimators that are more robust to MCAR may be out of scope, but it seems important to at least assess its impact on the estimation empirically. This seems possible, at least on the bladder data, or even on a separate simulation, eg by making the missing time sampling depend on the count. In the experiments, the estimator is compared to a naive baseline where missing values are set to zero, which strongly biases the estimation of the mean. Other baselines such as replacing missing values by the previous report or the median of all reports could be less biased, and it would be useful to include them in the comparison.

Correctness: The proposed method and analysis seem correct.

Clarity: The submission is very clear and well presented.

Relation to Prior Work: The relation to previous work is carefully discussed. The theoretical properties that are proved for the proposed non-parametric EM follow the line of recent parametric results. This extension seems non-trivial.

Reproducibility: Yes

Additional Feedback: I thank the authors for taking the time to do these additional experiments, they answer the minor points that I raised in my review.

Review 3

Summary and Contributions: The paper proposes an Expectation Maximization (EM) algorithm for panel count data that is censored through a Missing Completely At Random mechanism (MCAR).

Strengths: - The setup and problem description is well written. I enjoyed the description of the problem, particularly the description provided of Ecological Momentary Assessment and relating it to the panel count and missing data problem. - The assumptions are clearly stated which makes it easy to determine which parts of the methodology need to be carefully examined and could be improved in future works or conveyed to collaborators before they decide to use the method. - Principled methods for handling missing data are important.

Weaknesses: - The MCAR assumption is difficult to justify in practice. - Related to the MCAR assumption: Assumption 8 naturally states that the probability of missingness \epsilon is > 0 and from the definition earlier in the paper (it may be better to restate this bound in the assumptions section) that it is also strictly less than 1. - The bound 0 < \epsilon < 1 and MCAR together imply that any parameter of interest in the full law (that can be expressed as a function of the full law) is identified. This is good, however, could the authors clarify some of the following points regarding their method in the context of MCAR missingness. (i) Recovery of functions of the full law under MCAR missingness is very simple. By definition, MCAR implies that one can simply ignore any rows of data containing missingness and restricting the analysis to so called "complete cases" will still result in unbiased estimates of the parameter of interest. In light of this, and the bounds on \epsilon implying that there will always be complete cases in the data as n -> \infty (if this were not true, the parameters of interest would not be identifiable) what is the advantage of the proposed EM algorithm over simply doing complete case analysis and using some of the older tools cited in the paper that can be run on complete data. I apologize if I missed this, but it doesn't seem like there's a baseline comparison to such a complete case analysis or to the alternative of directly maximizing the observed data likelihood by integrating according to patterns of missingness. The latter yields the most (statistically) efficient estimates of the parameter of interest if the parametric form is correctly specified. I think both of these techniques are important baseline comparisons to discuss in theory as well as empirically. (ii) Related to the above point, it may be that in any finite sample there are no complete cases. Is it the case then that the EM proposal is meant to do the "best you can with finite samples"? The sample theory in section 5.4 seems to imply that n must still be "large enough". In which case, standard asymptotics from MCAR complete case analysis and direct maximization of the observed data likelihood should also apply. That is, both of these yield unbiased estimates with the latter being the most efficient way of obtaining the estimates. - I'm also unsure why the work needs to be restricted to MCAR mechanisms, and actually think it should be extensible to more complicated mechanisms without (much) change. That is, I don't think there are any portions of the proofs in the Appendix that would fail had it been a more complicated mechanism such that the full law is still recoverable from the observed data (however, the appendix is fairly lengthy and I may have missed some things). EM has been applied to many MAR problems and actually works for several MNAR mechanisms as long as the assumptions on the missingness process are clearly stated and these assumptions yield identified full laws. Examples of such works are from the graphical modeling literature of missing data processes listed below. [1] Full Law Identification in Graphical Models of Missing Data: Completeness Results (ICML 2020) [2] Identification In Missing Data Models Represented By Directed Acyclic Graphs (UAI 2019) [3] Graphical Models for Inference with Missing Data (NeurIPS 2013) The relevant point is that in these works, the missingness mechanism (MCAR, MAR, or MNAR) is modeled explicitly and if the proposed missingness mechanism implies that the full law is identified (see [1] for details), this means that maximization of the observed data likelihood (either through EM or by direct maximization of the observed data likelihood) will recover the true parameter of interest. So one nice way of extending the present work may be to posit a set of assumptions on the missingness mechanism using a graph (the underlying distribution need not be graphical, just the missingness process) as in the works listed above or by positing a set of algebraic assumptions that yield identification but these are often harder to convey to clinical collaborators. This is also probably why in D.5 of the Appendix the authors' experiments with MAR mechanisms still yield good results for recovery of the parameters of interest because when the data are MAR, the full law is always identifiable as a function of the observed law. For MNAR mechanisms however, one needs to be more careful as discussed in the works listed above and designing an EM procedure does get a bit more complicated because the identifying functionals are also more complicated. - I think the authors should be more ambitious and extend their methods beyond the MCAR setting because of the reasons stated above (complete case analysis being sufficient for MCAR and the present work possibly being extensible to identifiable MAR/MNAR without much change). This may require significant rewriting though and that plays a major part in my final score because conferences don't have a major revisions option. I think the paper will be quite strong after revisions to incorporate more complicated missingness mechanisms.

Correctness: - I did my best to check the claims and they seem fine to me.

Clarity: - As mentioned above I thought the introduction was quite well written. - Some other parts of the paper could do with reorganization though. There are still definitions being introduced towards the end of the paper. Of course this is a difficult balancing act given the 8 page format but I think the authors could consider restructuring so that technical definitions and intro are done by page 4/5 and there's more space given to intuitions for the proofs as the entirety of these proofs are currently restricted to a fairly lengthy supplement.

Relation to Prior Work: - Some of the work regarding results on the EM algorithm outside of the missingness component were a little confusing in terms of where the authors work began and where previous works ended. It may be good to state more explicitly what has already been shown and which specific portions are new in this paper. - Outside of the works I listed above, there are some other works in the missing data literature that may be useful to mention: [4] Block-Conditional Missing at Random Models for Missing Data (Statistical Science 2010) [5] Itemwise conditionally independent nonresponse modeling for incomplete multivariate data (Biometrika 2017) [6] Semiparametric Inference for Non-monotone Missing-Not-at-Random Data: the No Self-Censoring Model

Reproducibility: Yes

Additional Feedback: - The citations need some fixing. In many places EM is not capitalized appropriately. - While I appreciate the authors' honesty in their usage of MathStackExchange, it may be better to formalize the citation by referring to the original sources found in the StackExchange post i.e. references to the mean value theorem and Darboux's theorem and explicitly invoking these theorems instead of saying "which we can do by [9]" in the proof of Claim 2 where [9] simply refers to the MathStackExchange post. I may even suggest adding the reference to both the theorems as well as the MathStackExchange post. Additional references never hurt! ***************** Feedback after authors response and reviewer discussion ***************** - I have updated my score to a 5 to reflect that I appreciate the theoretical contributions to nonparametric finite sample EM theory. - However, I agree with other reviewers that the contribution in terms of missing data remains minimal if the theory can only be extended to MCAR data at present. - With regards to this, I believe the paper needs a fair amount of revision outside of the scope of this review process in order to redirect emphasis to results that are not currently reflected as being the primary contribution. That is, as it stands, the proposal is presented as a solution to missing data problems. However, it may be better to revise the manuscript to emphasize the nonparametric EM theory as a stand alone component, with applications to missing data as a specific use case. - Other points that I don't think were clarified fully in the author response include maximization of the observed data likelihood. I did not really grasp what the implications of the authors stating that there is no unique maximizer meant. If parameters of a model are identified, then there should be a unique global maximizer. If the parameters are not identified, then EM or any other technique will also face the same issues. That is, I think if there is no information in the likelihood about the parameter of interest, EM and other techniques cannot help learn the parameter. I think the authors should discuss identification in a little more detail.

Review 4

Summary and Contributions: The authors proposed a functional EM algorithm to estimate the mean function for incomplete panel count data. By extending EM algorithm to non-parametric settings, the authors provided finite sample convergence guarantees.

Strengths: Missing value imputation is a key problem in many areas (e.g., healthcare). For this reason, the problem discussed in this paper is very important. The theoretical convergence guarantees are quite interesting, although some key assumptions (e.g., MCAR) are strong and usually do not hold in reality.

Weaknesses: Besides the concerns about the assumptions (discussed in Strengths), my other major comment is related to experiments. 1. It seems that the proposed approach was not compared against any existing imputation method. Without doing so, it is very difficult to see the real value of the work. 2. In Section 6.1, the authors "artificially delete intervals completely at random with probability 0.2.". I am wondering how 0.2 was chosen? Based on my experiences, the probability could be much higher than 0.2 in reality. How does the proposed method work when it is the case? Some parameter analysis (the robustness of the work with respect to the probability) would be nice. 3. I find the results in Section 6.2 a bit weak. Do the results echo findings reported in the literature? Such comparisons could be useful.

Correctness: The theoretical analysis seems to be correct, although I did not investigate this thoroughly.

Clarity: The paper is quite well written.

Relation to Prior Work: It seems that important prior work have been discussed.

Reproducibility: Yes

Additional Feedback: While I appreciate the authors' feedback (adding experiments to address MNAR and to compare with rival methods), I feel that a theoretical analysis of how the proposed method performs under MNAR is lacking. Moreover, as I mentioned in the earlier review, a parameter analysis is necessary to evaluate the robustness of the method. For the above reasons, I would keep my score (marginally below the threshold of acceptance) unchanged.