(iii) Why causal adjustments are indispensable for causal inference tasks such as ours. Importantly, even though we agree that the strength of time-varying confounding is generally unknown, it would be irresponsible from a clinical perspective to not adjust for this source of bias. If we do not leverage proper adjustments for confounding, our method would always target an incorrect estimand. Estimating potential outcomes from observational data such as EHRs is always confounded, as treatment assignments depend on the current health condition of the patient, age, sex, social background, etc.

Hence, even though we do not know how much we gain from adjusting, we cannot ignore the existence of confounding in medical scenarios. Hence, our task requires a method that performs causal adjustments, and we therefore develop a causal method for this purpose.

(iv) Why oversampling does not help in our setting. Finally, we acknowledge that there is a connection between inverse propensity weighting and oversampling proportional to the inverse propensity weights. However, in real-world observational data, the propensity score is always unknown. Hence, there is simply no access to a ground truth reweighting / oversampling factor that one could use for generating true counterfactual outcomes solely based on observational data.

The above is commonly referred to as the fundamental problem of causal inference and is the reason why validating our results with real-world data is not possible. In other words, the interventional distribution is a counterfactual quantity. Hence, the standard approach in the literature is to validate causal ML models on synthetic data, which is consistent with the literature [5,6,7,8,9].

Action: In our updated version of the paper, we add a new Supplement H, where we provide intuition for causal inference in the time-varying setting and for adjustments on causal graphs (see new Figures 4 and 5). Thereby, we explain why standard, non-causal irregular time series methods fail to acknowledge the causal structure of our task and are thus inappropriate.

b. Differences to the literature on irregular time series

Thank you for this suggestion! We are happy to expand our literature review toward irregular time series (see our revised extended related work in Supplement A). If you have any specific literature in mind, we are happy to include this as well. Therein, we also discuss that traditional methods for irregular time series are different from our work. To this end, we would like to emphasize that, while our method operates within the irregular time series / continuous time setting, its main contribution is in the field of causal ML, as we are the first to propose a neural method that correctly adjusts for time-varying confounding in continuous time. Hence, while general continuous/irregular time literature is somewhat related, it aims at a different purpose.

Action: In our updated version, we added literature on irregular time series in our extended related work (see our new Supplement A). Therein, we also spell out clearly how our work is different and thus novel.

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2. Better Writing

Thank you for your comment!

Action: In our updated version, we removed the “=> Takeaway”, we rephrased L272, and we added more details of the experimental setup (e.g., meaning of the confounding parameter, what the RMSE refers to) in our main paper.

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Thank you very much for your helpful review!

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Responses to Weaknesses:

1. Why our method is different from traditional irregular time series prediction and why our setting is non-trivial.

Below, we would like to clarify why our setting is different from traditional irregular time series prediction and why our setting is challenging. Upon reading your questions, we realized that we should have done a better job in connecting our work to irregular time series literature and thereby explaining why our method is different and thus novel. We thus added a new section where we introduce the additional challenges due to our causal task (see our new Supplement H), and provide intuition on a causal graph (see our new Figures 4 and 5) . Below, we provide a detailed step-by-step answer.

a. Adjustments for time-varying confounding and use of synthetic data

Thank you for your comment! Upon reading your comment, we realized that there might be some confusion. In the following, we would like to clarify what (i) oversampling is typically used for, (ii) why we need adjustments for time-varying confounders, (iii) why adjustments are indispensable for estimating potential outcomes, and (iv) why oversampling does not help in our setting. Finally, we also explain the challenges when evaluating counterfactuals.

(i) Purpose of oversampling. Oversampling is a useful technique when there is a target imbalance in the training data. It is commonly used to reduce finite sample estimation bias in standard regression tasks. That is, when outcomes are imbalanced, training a machine learning model on such a dataset will result in a biased estimator. If the task at hand is a regression or a classification, performing oversampling will help mitigate this finite sample estimation bias.

(ii) Why we need adjustments for time-varying confounders (and not oversampling). The above, however, is not the problem in our setting. Instead, the main motivation of our work is to present a computationally tractable, correct estimand for the potential outcome (which is a causal quantity and requires so-called causal adjustments). Hence, we need inverse propensity weights to target the correct estimand. Thus, this has a purpose that is different to oversampling. These inverse propensity weights are, however, unknown in observational studies.

To be more explicit, let’s consider the following scenario: Assume we had infinite data. Then, if our task was to perform a simple regression $E[Y|X=x]$, we could fit an arbitrary regression model. If our model capacity is great enough, our estimation bias goes to 0 when there is infinite data.

However, even if we had infinite data, performing a standard regression (as in traditional irregular time series prediction) for estimating potential outcomes over time would be biased, as we would target an incorrect estimand. Formally, the potential outcome does not coincide with the conditional expectation [1]:

$\mathbb{E}[Y_\tau [a_{t:\tau}] | H_t=h_t] \neq \mathbb{E}[Y_\tau | H_t=h_t, A_{t:\tau}=a_{t:\tau}].$

The problem lies in the sequential nature of the estimation task. During inference, only some of the future confounders are observed, which is also known as runtime confounding in the static setting [2]. In particular, for treatments $A_{t+\delta}$, only the confounders that lie in the history $H_t$ are observed, but not in the interval $(t,t+\delta)$. Hence, causal adjustments for these confounders are required, or, otherwise, we will always suffer from bias due to not targeting the correct estimand. In other words, traditional irregular time series methods do not make such causal adjustments and therefore target an incorrect estimand. This will lead to bias, irrespective of the amount of training data. Instead, our method targets the correct estimand.

Motivated by the above, we need for our causal inference setting a tractable, alternative expression for the potential outcome (the left hand side of the above equation). This is a well-studied problem in the causal inference and statistics literature and, for discrete time, dates back to works such as [3,4].

One possible way to reformulate the potential outcome (i.e., the estimand on the left hand side) is via inverse propensity weighting. While inverse propensity has already been employed by neural networks such as [5] in the discrete-time setting, there is no work that has derived tractable inverse propensity weights in continuous / irregular time. This is one of the main novelties of our work.

Thank you for the detailed response. However, I think you might have misread my first comment.

My comment was about potential flaws in the evaluation, which in turn weakens the motivation. My main concern is that the added complexity about confoundings cannot be shown to be better than conceptually simpler alternatives (that directly predicts Y from X without considering any confounding variables). Thus, I mentioned oversampling as a way to potentially show your method is better than others. Currently this is shown via simulations which is unconvincing. One big benefit of good causal inference models is that it can be more robust to spurious correlation, so if the proposed method is actually better, I'd imagine it should be more robust to covariate shift (via oversampling) as well.

(This is related to the baseline choice as well) I understand that the authors might be coming from a causal inference background, but currently the motivation is essentially "we have to use causal inference in this task", which is weak if we cannot show that it also improve the performance on the task, at least in some dimensions. For this reason, I was also hoping there would more competitive non-causal-inference-related baselines.

Also, for my question on confounding strength, by "what $\gamma$ means" I mean its relationship to real-world quantity. For example, can we know what's a realistic $\gamma$ (1? 100?), given a particular prediction task based on the MIMIC-III dataset?

Thank you for your comment! Please let us clarify the motivation of our work and our experiments:

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Evaluation:

The task that our method aims to solve is a causal inference task, that is, estimating a counterfactual quantity. In order to correctly evaluate a method for counterfactual estimation, we need access to true counterfactual quantities. These are, however, fundamentally unobserved – which is the fundamental problem of causal inference [1]. Please note that the task of traditional ML lies on a lower rung of Pearl’s ladder of causation than our task. We have an introduction and a discussion on this in our Supplement H.

Generating true counterfactuals from observational data, for example via oversampling, is impossible. We would need to oversample quantities proportional to the true inverse propensity score. The true propensity score in observational data is, however, fundamentally unknown. This is the difference between observational data and randomized control trials (RCTs).

If there was a way to generate true counterfactuals from observational data – say, via oversampling – this would imply that

• (i) the fundamental problem of causal inference was solved and, hence, the whole literature on treatment effect estimation and potential outcome estimation was obsolete, and
• (ii) there would be no point in making randomized control trials, as we could generate any true counterfactual outcome out of thin air.

Therefore, evaluating our method and baselines on (semi-)synthetic data is the only valid way and is consistent with the literature.

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Baselines:

Regarding baselines, benchmarking our method with existing baselines for our task is the only fair comparison. Nevertheless, we are more than happy to provide further experimental results to show this empirically. Hence, to demonstrate that no-adjustment methods are insufficient for our task, we added a neural CDE baseline that directly fits the observed data. We see that the new baseline is outperformed by our proposed method by a margin.

Action: We added the additional experiments with the new neural CDE baseline ("No-adjustment CDE") to our experiments in Section 5.

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Confounding strength:

The issue that the true confounding strength $\gamma$ is unknown is exactly our point: we do not know it, therefore, it is inappropriate to simply ignore this quantity. Applying simple no-adjustment methods in clinical scenarios would be completely irresponsible, as we would know that we target an incorrect estimand and are always biased.

Ignoring the existence of confounding would further mean that RCT studies are purposeless.

Again, if we could perform simple regressions on observational data for our task, this would mean that we could infer counterfactual quantities from observational data without adjustments. Therefore, both (i) causal inference on observational data, and (ii) RCT studies would be completely obsolete.

Action: We added a new no-adjustment baseline to our evaluation ("No-adjustment CDE"). The results demonstrate that applying no adjustments for counterfactual outcome estimation leads to very poor predictions, as it is inappropriate for our task. Again, this confirms the significance of our proposed method.

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[1] Guido W. Imbens and Donald B. Rubin. Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge books online. Cambridge University Press, Cambridge, 2015.

Thank you very much for your positive review of our paper!

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Responses to Weaknesses:

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1. Further details about data-generating process

Thank you. We are provided details regarding our data-generating process (see our Supplement E.1). Therein, we detailed that the data-generating process in [1] has two main differences to our DGP. We emphasize that our choices are consistent with the literature:

(i) The work in [1] uses an RCT style setting where treatments are assigned fully randomized. Hence, there is no time-varying confounding. In contrast, our work is interested in a setup with time-varying confounding. Hence, we added time-varying confounding, which is consistent with prior literature [2,3,4,5]. In particular, our treatment assignment process follows Eq. (137), which is consistent with [4].

(ii) The work in [1] analyzes the informativeness of covariate/outcome observation times. This is not the focus of our work. Hence, we set the informativeness parameter in Eq. (138) to $\omega=0$ (in [1], $\omega$ is referred to as $\gamma$ in their Eq. (7)).

Action: We discuss the choice of our data-generating process with greater care and move important details to our revised Section 5.

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2. Distribution of timestamps / regular timestamps

Thank you for this suggestion! Analyzing the impact of different irregularities in the sampling times is a great idea! We are happy to offer new experiments. In our updated version, we added two new experimental setups in Supplements F.1 and F.2:

(i) We analyze how our SCIP-Net performs under different sampling distributions. For this, we vary the aforementioned informativeness parameter of the sampling times $\omega$.

(ii) We add an additional ablation, which we call the SDIP-Net. Therein, we show that our stabilized IPW approach is also applicable to the (more unrealistic) discrete time scenario. For this, we use a simple LSTM as a neural backbone, which further shows that our framework is applicable to different neural architectures. We find that our ablation has comparable performance to discrete-time baselines (yet we emphasize that our contribution is to focus on the continuous-time setting and not the discrete-time setting).

Both experiments again confirm the effectiveness of our proposed stabilized IPW.

Action: We added the new experiments for different irregularities as well as an ablation study for discrete time to a new Supplement F. The results demonstrate again the effectiveness of our proposed stabilized IPW.

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Responses to Questions:

1. Font size

Thank you for pointing this out.

Action: In our updated version, we increased the font size.

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2. Difference in performance of TE-CDE

Thank you for this question. We have carefully looked into the underlying reasons, and, hence, see the reason in the differences in the data generating process. In TE-CDE [5], the data-generating process is based on the Hawkes process for masking observation times, which is re-run 5 times and the resulting observation masks are re-applied on the timeseries each time. Hence, the final observation times are 5x sparser than actually specified by the chosen hyperparameters (in their code repository, this can be found in TE-CDE/src/utils/process_irregular_data.py, lines 286-289). We did not find an explanation for this choice, nor do we think it is a reasonable choice to mask away that many observations without further clarification. Hence, we opted for the simulation setup for irregular timestamps as in [1] (with sampling informativeness set to $\omega=0$).

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[1] Toon Vanderschueren, Alicia Curth, Wouter Verbeke, and Mihaela van der Schaar. Accounting for informative sampling when learning to forecast treatment outcomes over time. In ICML, 2023.

[2] Ioana Bica, Ahmed M. Alaa, James Jordon, and Mihaela van der Schaar. Estimating counterfactual treatment outcomes over time through adversarially balanced representations. In ICLR, 2020.

[3] Bryan Lim, Ahmed M. Alaa, and Mihaela van der Schaar. Forecasting treatment responses over time using recurrent marginal structural networks. In NeurIPS, 2018.

[4] Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel. Causal transformer for estimating counterfactual outcomes. In ICML, 2022.

[5] Nabeel Seedat, Fergus Imrie, Alexis Bellot, Zhaozhi Qian, and Mihaela van der Schaar. Continuous-time modeling of counterfactual outcomes using neural controlled differential equations. In ICML, 2022.

Thank you for the updates and additional details.

While Supplementary F provides valuable information, my question remains unresolved regarding why SCIP-Net and its ablation without stabilized weights (CIP-Net) did not exhibit a significant performance difference in experiments when the prediction horizon is small. Could the authors share any idea to clarify this observation?

Minor thing: The first column heading in Table 4,5 in Supplementary F should be informativenessof thesampling? not confounding strength?

Dear reviewer bgav,

We thank you for your helpful feedback on our submission.

We are happy that you appreciate our additional experimental results, and we hope that our explanation on the similar performance of SCIP-Net and its ablation for small prediction horizons could clarify your remaining question. Otherwise, we are happy to provide further insights!

If you feel comfortable with it, we would highly appreciate it if you would consider increasing your score.

Third, let’s move to our time-varying setting.

In our time-varying setting, we want to ensure that we target the correct estimand, for which we employ inverse propensity weighting. It is unique to the time-varying setting that a simple backdoor-adjustment is not sufficient to adjust for time-varying confounding [8]. That is, $\mathbb{E}[Y_\tau [a_{t:\tau}] | H_t=h_t] \neq \mathbb{E}[Y_\tau | H_t=h_t, A_{t:\tau}=a_{t:\tau}].$

Methods such as [1], [9], [10] use a balancing objective on top of performing simple backdoor adjustments $\mathbb{E}[Y_\tau | H_t=h_t, A_{t:\tau}=a_{t:\tau}]$. For this, they employ adversarial training losses of the form $\text{MSE} - \lambda \times\text{Balancing loss}.$

Clearly, even for infinite data, the adversarial training objective in these methods depends on a balancing parameter $\lambda$. Hence, these methods target an estimand $\mathbb{E}^\lambda[(\cdot) | H_t=h_t, A_{t:\tau}=a_{t:\tau}]$ that also depends on a balancing hyperparameter $\lambda$. Here, $(\cdot)$ stands for an unknown quantity, as in the adversarial objective, it is unclear what the targeted criterion is for fixed $\lambda$.

However, in general, there does not exist a balancing hyperparameter $\lambda$ such that the estimand $\mathbb{E}^\lambda[(\cdot) | H_t=h_t, A_{t:\tau}=a_{t:\tau}]$ coincides with the conditional average potential outcome $\mathbb{E}[Y_\tau [a_{t:\tau}] | H_t=h_t].$

And even if there was such a $\lambda$, there would be no way to validate the choice on observational data.

The improved performance due to balancing in works for the time-varying setting like [1,9,10] is most likely due to a reduction in finite-sample estimation variance. Further, the proofs in [9] and [10] guarantee only the following: under certain conditions, the minimax game induced by the adversarial loss has a global minimum, which is attained for representations that are invariant to the treatment assigned. This, however, by no means implies that the correct estimand is targeted.

Instead, in order to target $\mathbb{E}[Y_\tau [a_{t:\tau}] | H_t=h_t]$, proper adjustments for time-varying confounding such as G-computation or inverse propensity weighting [11,12] are needed.

Finally, we corrected the reference to state that balancing may increase bias (it was not Melnychuk et al. [2] but Melnychuk et al. [13]).

Action: We now cite [4] and [13] properly. We further spell out more clearly and with greater detail why balancing aims at reducing estimation variance in our setting and is not a proper adjustment. For this, we provide an intuition for adjustments for time-varying confounders (see our new Supplement H) through additional figures (see our new Figure 5), and clarified the purpose of balancing in the causal inference literature (see our new Supplement I).

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Clarification: Potential outcome vs Conditional average potential outcome

Thank you for pointing this out. You are correct in that our target is the conditional average potential outcome. Proposition 1, indeed, relates to the CAPO. We chose the title of our work for brevity and readability, but we are happy to change it if you prefer.

Action: In our updated version, we use the proper conditional average potential outcome (CAPO) terminology throughout the main body of the paper. In particular, we have now revised Proposition 1 as per your suggestion.

Thank you very much for your positive review of our paper!

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Responses to Weaknesses:

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W1. Non-trivial contribution

Thank you for your comment. It is correct that (i) TE-CDE [1] also leverages neural CDEs [14] for predicting potential outcomes in continuous time, and that (ii) we modeled our likelihood akin to Rytgaard et al. [3]. However, our choice for neural CDEs and product integrals are deliberate: (i) neural CDEs are the state-of-the-art neural architecture for continuous time estimation tasks. (ii) Our likelihood closely follows the statistics literature via product integrals, as the mathematical tools have carefully been developed there. We think that this is an advantage of our work, as we leverage existing literature.

However, we contribute in three different ways: We are the first to derive a (i) tractable IPW objective in continuous time that can be estimated with neural networks. Additionally, we are the first to develop (ii) stabilized inverse propensity weights in continuous time. Finally, we develop the (iii) first neural instantiation that adjusts for time-varying confounding in continuous time.

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W2. Balancing

Thank you for your comment! We are happy to offer a more detailed explanation of the purpose of balancing.

First, you are correct: balancing was originally proposed in [4] for counterfactual inference in the static setting. We did not properly cite this work but only the follow-up work [5]. We apologize for this! We now properly cite [4] .

Second, we are happy to clarify the purpose of balancing in paper [4] and related literature. In the following, we first discuss the paper [4] and, after that, discuss the implications for our time-varying setting.

In [4], the authors work with the “Rubin-Neyman causal model” [6,7] and, hence, operate under the three standard assumptions of (i) consistency, (ii) overlap, and (iii) ignorability. These are sufficient conditions for the backdoor adjustment to adjust for all confounders. Hence, under this model, they target the correct estimand without suffering from confounding bias. Hence, balancing is not needed to address confounding bias.

Rather, the goal of balancing in [4] is to reduce estimation variance due to the distribution shift from observational to interventional/counterfactual distribution. Say, for example, treatment $A=a_*$ is the treatment of interest for counterfactual inference. However, given certain covariate values, this treatment is barely recorded in the observational dataset. Hence, the counterfactual outcome can only be estimated with large variance due to low overlap in the training set.

Indeed, the authors of [4] suggested balancing to reduce this finite-sample estimation variance. This is explicitly written in their work [4]: “We then introduce a form of regularization by enforcing similarity between the distributions of representations learned for populations with different interventions. (...) This reduces the variance from fitting a model on one distribution and applying it to another.”

In the setting from [4], balancing is not required to ensure that they target the correct estimand (that is, avoid infinite data bias). If the method in [4] had infinite data, the true causal parameter could be recovered, even without balancing. The authors in [4] thus only employ balancing to reduce the variance of their finite-sample estimators.

Dear reviewer utuj,

We thank you for your helpful feedback on our submission. We hope that we could address your questions, and we are happy to provide more clarification otherwise.

If you feel comfortable with it, we would highly appreciate it if you would consider increasing your score.

Thank you very much for your positive review of our paper!

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Responses to Weaknesses & Questions

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1. Details on difficulties during training

Thank you for pointing this out. We found that, overall, our training was fairly robust, and we generally did not experience any particular difficulties. Nevertheless, we wanted to share one important ‘lesson learned’ that we found important for achieving robustness in our experiment and which we think is relevant for potential users of our method. Here, we found that the choice of the numerical integration is crucial to balance a low runtime and a stable performance. The reasons are the following: Different to LSTM and transformer-based methods, our SCIP-Net requires numerical quadrature schemes for (i) solving the neural CDEs and (ii) integrating the treatment intensity (i.e., the numerator in the inverse propensity weight and the denominator in the scaling factor). This may lead to an additional computational overhead and can be quite involved when using higher-order quadrature schemes such as Runge-Kutta. Surprisingly, we found that Euler integration worked very well and reduced computational complexity considerably without significant approximation errors. Hence, for our experiments, we did not need to resort to more complex integration schemes and thus recommend the use of Euler integration in practice.

Action: In our updated paper, we added details on the numerical quadrature scheme to the implementation details (Supplement G) for both our SCIP-Net (and TE-CDE [2]), and provided further comments on the choice.

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[1] Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel. Causal transformer for estimating counterfactual outcomes. In ICML, 2022.

[2] Nabeel Seedat, Fergus Imrie, Alexis Bellot, Zhaozhi Qian, and Mihaela van der Schaar. Continuous-time modeling of counterfactual outcomes using neural controlled differential equations. In ICML, 2022.