Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation

Thank you very much for the constructive evaluation of our paper. We are happy to address your questions below.

# Response to “Weaknesses”

Response to (1a) neural SDEs and neural CDEs:

Thank you for giving us the opportunity to clarify the novelty of our method and how we add to the existing literature. In particular, our work makes non-trivial contributions along three literature streams.

1. We contribute to the literature on neural differential equations. Neural CDE [1] and neural SDE [2] have been proposed previously in isolation, while ours is the first method to combine them into a single method. Further, we are not aware of any prior work that has coupled neural SDEs in an encoder-decoder framework. More importantly, to estimate treatment effects, a simple ‘plug-n-play’ approach combining both is typically not sufficient. In contrast, our method extends them in a non-trivial way with additional techniques from the causal machine learning literature (e.g., informative sampling in Supplement L, balanced representations in new Supplement M). Moreover, there does not exist a single tailored Bayesian neural CDE method.
2. We contribute to the literature on treatment effect estimation. Existing methods for treatment effect estimation in continuous time (TE-CDE [3]) are limited to point estimates. So far, no method for uncertainty quantification in this setting existed. Ours is the first method that allows for rigorous, Bayesian uncertainty quantification.
3. We contribute to the medical literature. We show that the MC dropout [4] (as used in our baselines [3]) gives incorrect uncertainty estimates, which may lead to harmful decisions in medical practice. As a remedy, we offer a new method to support reliable medical decision-making over time by providing reliable uncertainty estimates. As such, we study an important and underexplored problem in medicine.

Action: We carefully checked that we clearly spelled out the novelty of our work. In particular, we revised Section 1 and emphasized our contribution to the three different literature streams.

Response to (1b) distinction from CF-ODE:

Thank you for giving us the opportunity to highlight the differences between our BNCDE and the CF-ODE [5]. The are six important differences:

1. CF-ODE is inherently designed to forecast the treatment effect over time of a single, static treatment. In contrast, our BNCDE is designed to estimate treatment effects for a sequence of multiple treatments in the future. This is a much more complex task.
2. Accordingly, CF-ODE builds upon a static identifiability framework. Here, adjusting for the measurements of the patient history $S_{t^*}(X)$ at the time of treatment, $t^*$ is assumed to be sufficient to adjust for all confounders (Assumption 2.3).
3. CF-ODE directly models the latent states in the decoder with a neural SDE. We do not see how this can be effectively combined with neural CDEs to model arbitrary, future sequences of multiple treatments.
4. All patient information in CF-ODE is contained deterministically in the initial state of the ODE. Hence, all uncertainty needs to be incorporated in a deterministic, initial state. We argue that this is very inefficient. Instead, our BNCDE captures patient information in the random variable $Z_T$. Here, all the Monte Carlo variance is propagated through the encoded trajectory $Z_{[0,T]}$, accounting for uncertainty at each point in time $t \in [0,T]$. Therefore, model uncertainty is directly incorporated into the patient encoding.
5. The use of a GRU in the CF-ODE encoder implies regular sampling of the patient trajectories. This assumption is a crucial violation of medical reality, where it is standard that measurements are taken at irregular points in time. Therefore, CF-ODE does not account for uncertainty due to irregular sampling.
6. The single treatment decision in CF-ODE is limited to binary treatments. Instead, our BNCDE can deal with any treatment (in our experiments, treatments are discrete).

Because of the above, CF-ODE is not applicable to our setting.

Action: We added a new Supplement F where we spell out the differences between CF-ODE and our method. In sum, CF-ODE is not applicable to our setting. We also refer to our new Supplement E, where we further highlight the limitations of neural ODE modeling.

Response to (2) scalability:

This is an excellent question. Increasing the depth of the neural CDE networks directly affects the dimension of the latent neural SDEs. Accordingly, the differential equation solver must solve a higher dimensional SDE. Dimensionality is, however, not a real problem for our method for several reasons:

• (i) In Supplement I, we report the average training time of our method. Overall, the average time of our BNCDE was comparable to the TE-CDE [3] baseline (approx. 34h vs 22h).
• (ii) We provide new experiments in Supplement N which show that the training time increases only linearly with the depth of the neural CDE networks (i.e., the dimension of the neural SDE). Linear scaling is not different from any other, fully connected neural network architecture.
• (iii) Furthermore, using multiple Monte Carlo trajectories at the same time reduces Monte Carlo variance during the training and, thus, stabilizes training. In the new Supplement N, we show that increasing the number of Monte Carlo samples comes at very little extra cost.
• (iv) We did not encounter any instability issues for our method. Our results are very consistent over different seeds (see our experiments in Section 5).
• (v) Importantly, discretization errors in SDE solvers do not increase with the dimension, but only depend on the order of the solver and the chosen time grid. Modern solvers can adaptively choose the grid size and provide error guarantees.

Action: We added new experiments in Supplement N, where we show that the training time only increases linearly with the depth of the neural CDEs, i.e., dimension of the neural SDEs. Further, we demonstrate that using multiple Monte Carlo samples at the same time comes at very little extra cost.

Thank you very much. We understand that scalability is an important issue. Therefore, we updated our work according to your suggestions. We highlighted in Section 4.2 that our work focuses on variational inference for neural CDEs of moderate size, and that stability is unexplored for high dimensional settings. Further, we added a word of caution in Supplement N. Finally, we extended our discussion on the applicability in medical practice in Supplement A.

Response to (3): Scalability

Thank you for your question. We are happy to clarify why problems that arise with these types of networks are not problematic in our scenario. Importantly, our proposed method is carefully designed for uncertainty quantification in medical settings, where uncertainty estimates are of utmost importance for decision-making.

• We would highlight that uncertainty quantification is typically needed for moderate-sized (rather than large) datasets in practice. The reason is that, for moderate-sized datasets, we cannot expect to learn the true data-generating mechanisms. Instead, we expect that there is uncertainty in how medical treatments affect patient health and, to deal with this, it is imperative in medicine to make treatment choices by incorporating the underlying uncertainty [7].

• We emphasize that medical datasets (for which our method is designed for) typically have a moderate number of data points in the order ten thousand data points (e.g., [2,3,4,5,6]), making the use of heavily overparameterized neural networks with multiple billions parameters often unnecessary or even infeasible. Even more, using a network architecture with multiple billions of parameters would overfit on datasets from medicine, which would completely contradict the task of our method, which is reliable uncertainty quantification.

• Of note, even transformers for healthcare applications also limit the number of parameters and do not use multiple billions of parameters, thereby avoiding the aforementioned overparameterization [9, 10]. Further, we emphasize that neural CDEs typically require fewer parameters than transformers, which have a much more complex architecture.

To address your point, we also show in our simulations (see Supplement N) that reducing Monte Carlo variance in our training comes at basically no cost. Similar to other neural architectures, training time only increases linearly with the depth of the networks. That is, it increases with $\mathcal{O}(d_\omega)$ in the notation from our paper. Hence, there are no drawbacks during training that are specific to our method.

Finally, we reported all of our results in the main section over 5 different seeds. Here, we see that our method produced highly stable results over all seeds.

[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] Alistair E. W. Johnson, Tom J. Pollard, Lu Shen, Li-wei H. Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G. Mark. MIMIC-III, a freely accessible critical care database. Scientific Data, 3(1):160035, 2016.

[3] Tom J. Pollard, Alistair E. W. Johnson, Jesse D. Raffa, Leo A. Celi, Roger G. Mark, and Omar Badawi. The eICU Collaborative Research Database, a freely available multi-center database for critical care research. Scientific Data, 5:180178, 2018.

[4] Stephanie L. Hyland, Martin Faltys, Matthias H¨user, Xinrui Lyu, Thomas Gumbsch, Cristobal Esteban, Christian Bock, Max Horn, Michael Moor, Bastian Rieck, Marc Zimmermann, Dean Bodenham, Karsten Borgwardt, Gunnar R¨atsch, and Tobias M. Merz. Early prediction of circulatory failure in the intensive care unit using machine learning. Nature medicine, 26(3):364–373, 2020.

[5] Patrick J. Thoral, Jan M. Peppink, Ronald H. Driessen, Eric J. G. Sijbrands, Erwin J. O. Kompanje, Lewis Kaplan, Heatherlee Bailey, Jozef Kesecioglu, Maurizio Cecconi, Matthew Churpek, Gilles Clermont, Mihaela van der Schaar, Ari Ercole, Armand R. J. Girbes, and Paul W. G. Elbers. Sharing ICU patient data responsibly under the Society of Critical Care Medicine/European Society of Intensive Care Medicine Joint Data Science Collaboration: The Amsterdam University Medical Centers Database (AmsterdamUMCdb) example. Critical Care Medicine, 49(6):563–577, 2021.

[6] Niklas Rodemund, Bernhard Wernly, Christian Jung, Crispiana Cozowicz, and Andreas Koköfer. The Salzburg Intensive Care database (SICdb): an openly available critical care dataset. Intensive care medicine, 49(6):700–702, 2023.

[7] Christopher R. S. Banerji, Tapabrata Chakraborti, Chris Harbron, and Ben D. MacArthur. Clinical AI tools must convey predictive uncertainty for each individual patient. Nature medicine, 2023.

[8] Edward De Brouwer, Javier Gonzalez Hernandez, and Stephanie Hyland. Predicting the impact of treatments over time with uncertainty aware neural differential equations. In AISTATS, 2022.

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

[10] Emmi Antikainen, Joonas Linnosmaa, Adil Umer, Niku Oksala, Markku Eskola, Mark van Gils, Jussi Hernesniemi, and Moncef Gabbouj. Transformers for cardiac patient mortality risk prediction from heterogeneous electronic health records. Scientific Reports, 13(1):3517, 2023.

Thank you very much for your response. We appreciate your feedback and we apologize if we were not clear enough in our first response.

Response to (1): novelty of balanced representations and informative sampling

Thank you. We do not claim that balancing and informative sampling is our novelty. Rather, we would like to highlight that our proposed method is capable of incorporating existing contributions from the causal ML literature. As such, we see the ideas of balanced representations and informative sampling orthogonal to our main contribution, which is: Bayesian uncertainty quantification for treatment effect estimation in continuous time.

Importantly, the key contribution of our method is orthogonal to the above topics such as balancing in that our method can produce highly reliable estimates of uncertainty (both aleatoric and epistemic). Our method achieves all of the following in a single, neural end-to-end architecture. Specifically, our method generates

1. informative epistemic uncertainty quantification
2. and meaningful aleatoric uncertainty quantification of the potential outcome
3. for any kind of treatment (e.g., binary, discrete, …)
4. and multiple sequences of treatments in the future
5. in continuous time.

So far, there does not exist a single method that achieves the above. In addition, our method is also compatible with innovations from earlier research (i.e., balanced representations and informative sampling).

Response to (2): Non-applicability of CF-ODE

Thank you for your follow-up question. We apologize if we were not clear enough. CF-ODE is a method that is designed for the single time intervention setting. This is a completely different setting from our time-varying treatment effect estimation setting. Hence, it is not possible to adapt to it.

The technical challenges are located in the different settings:

• CF-ODE builds on a static identifiability framework. It is therefore completely designed to this specific setting. The mathematical framework of CF-ODE, i.e. the architecture, the identifiability assumption, the causal diagram, is completely designed for one, single binary treatment.

• Therefore, CF-ODE can only deal with a single, static treatment, and this is a choice by design (that one cannot circumvent). We emphasize that this is (1) a much easier task and (2) contradicts medical reality, where schedules of multiple treatments are planned for the future. In contrast, our method can deal with a sequence of discrete treatments in the future.

So, why is not possible to adapt CF-ODE from the static to the time-varying setting?

• In order to account for multiple treatments in the future, it would be necessary to incorporate, for example, a neural CDE into CF-ODE. Both are entirely different techniques. We provide a clear distinction between neural CDE and neural ODE in our new Supplement E.

• However, multiple treatments via neural CDEs are not compatible with CF-ODE. First, the mathematical derivations all build around the single, static treatment (see, for example, definitions of $h(t^*)$, $u_T(t-t*)$, and the causal graph (Figure 2) in Section 2 of CF-ODE [8]). It would therefore be necessary to develop a completely different mathematical framework, find different derivations and develop a completely different architecture to incorporate multiple treatments.

• Further, CF-ODE directly models the latent states in their architecture with a neural SDE, see Eq. (5) in Section 3 of CF-ODE [8]. Because of this, it is not possible to directly insert a CDE into the CF-ODE architecture. To address this in method, we design our latent states in a way they are captured with a neural CDE. Instead of directly modeling uncertainty of the latent states, we model uncertainty through the parameterization of the latent states. That is, we build upon a coupled system of neural CDEs and neural SDEs. In particular, such a method has never been proposed before.

• Further, we emphasize that there are direct limitations of how uncertainty is captured in the CF-ODE encoder, which is, deterministically and in discrete time. This imposes strong assumptions on medical reality.

Hence, we summarize: CF-ODE proceeds by (1) directly modeling latent states with a neural SDE and, therefore, (2) can not be combined with neural CDEs. Further, CF-ODE (3) builds upon a static framework, which (4) does not account for discrete treatments, and (5) completely hinders the possibility for multiple treatments in the future. All derivations and the architecture are designed for a single, static treatment. Finally, (6) it is very limited in capturing uncertainty through the patient input.

Therefore, CF-ODE is not at all compatible with our setting, and we do not see how it can be extended to our setting.

Thank you very much for your helpful feedback on our paper. We appreciate that you find our paper crucial, innovative, and unique.

Response to (1) overlap assumption & intensity process:

Thank you for your question. Upon reading your questions, we realized that we should have spelled out more explicitly how point processes are used. You are correct: the observation times of the covariates $X_t$ and outcomes $Y_t$ follow a point process with a, possibly history-dependent, intensity function, say $\zeta(t)$. However, the point process for the observation times is different from the point process in the identifiability assumption. The latter is a point process with intensity $\lambda(t | H_t)$, where $\lambda(t | H_t)$ is the intensity function that governs the treatment assignment. Intuitively, this can be thought of as the analog of the propensity score in the static setting. For any given time $t$, we want the probability of treatment to be greater $0$ to ensure identifiability. In the continuous-time setting, this means that the intensity function $\lambda(t|H_t)$ is non-zero. If the intensity function was hypothetically zero, then a patient with history $H_t$ would have had no opportunity to receive treatment at time $t$. Instead, if $\lambda(t|H_t)>0$, there is a chance that treatment is administered.

We can attempt to verify this assumption in practice empirically. In the static setting, one would simply estimate the propensity score from data (see, e.g., [1]). We can also adopt the same strategy in our continuous-time setting. Here, we can simply estimate the intensity function from data, for example, via maximum likelihood or nonparametric approaches [2]. Thereby, we can verify whether the intensity satisfies the overlap assumption.

Action: We clarified that the observation times of $X_t$ and $Y_t$ follow a point process $\zeta(t)$ that is different from the treatment assignment process $\lambda(t)$ (see our revised Section 3). We further explained how the treatment intensity can be estimated from data, so that the overlap assumption can be verified (see our revised Section 3).

Response to (2) posterior variance

Thank you for asking this important question. In theory, the variational weight posterior variance could also be smaller than the true weight posterior variance. In such cases, uncertainty estimates will be overconfident. This has two implications: if practitioners rely on uncertainty estimates as a decision rule to administer a treatment, they may be misled. Further, the posterior predictive distribution will become overconfident.

However, our empirical findings show that this is not an issue for our BNCDE. Across all experiments, we find that the posterior predictive distribution of our BNCDE tends to be conservative (see, e.g., Figure 2 and the additional studies in Supplements K, L, M). That is, it tends to provide more coverage than the predictive posterior credible intervals should guarantee from a frequentist point of view. Further, although we cannot assess the true, underlying posterior variance, we highlight that Figure 4 clearly shows the informativeness of our estimated weight posterior variance. That is, we observe meaningful differences in magnitude, which demonstrates the effectiveness of our method.

During hyperparameter tuning of our BNCDE, we further noticed that a higher diffusion hyperparameter in the latent SDEs generally led to more conservativeness in the potential outcome estimation. We therefore suggest that a careful choice of the diffusion hyperparameter is one of the most important hyperparameters in our method for applications in medical scenarios.

Action: We added the above recommendations for choosing the diffusion hyperparameter to our implementation details (see revised Supplement I). Furthermore, we discuss that a larger diffusion hyperparameter leads to more conservative estimates of the posterior distribution (see new Supplement A).

Thank you very much for your positive feedback on our paper! We are happy to answer your questions in the following.

Response to 6Zs2:

# Response to Weaknesses

Response to (1) possible downsides and robustness

Thank you. We added a new discussion around possible downsides and the assumptions that need to be satisfied to ensure robustness (see new Supplement A). Therein, we especially focus on potential limitations (e.g., explainability, scalability, etc.) that are relevant for medical practice.

Action: We added a new discussion on the applicability in medical practice (see new Supplement A).

Response to (2) comparison to non-neural methods

Thank you. We are more than happy to explain why non-neural methods are not applicable (see Table 1).

• To the best of our knowledge, the only applicable baseline (both neural and non-neural) is TE-CDE [1]. TE-CDE builds upon the same setting (i.e., treatment effect estimation in continuous time) as our work. However, it does not capture aleatoric and epistemic uncertainty. Hence, we extended TE-CDE with MC dropout [5] to obtain a baseline that is applicable to our setting.
• There are some non-neural methods that focus on a related setting [2,3,4] but still are not applicable. (We cited all of them in our related work section.) There are different reasons for that: they impose strong assumptions on the outcome distribution of the potential outcomes, they are not designed for multi-dimensional outcomes and static covariate data, and scalability is limited. For example, the methods in [4] can only deal with continuous treatments but not discrete treatments as in our work. The Gaussian process from [3] can not deal with static covariates because of which all patient-level information is ignored. Further, similar to [2], it scales cubically because of which practical application in medicine is prohibited.

Action: We have carefully checked our related work section to explain that existing, non-neural baselines are not applicable. Furthermore, we reiterate the downsides of existing baselines and the strengths of our proposed method in our new discussion (see new Supplement A).

# Responses to Questions

Response to (1) clarification of robustness

Thank you. Upon reading your comment, we found that we should remove “fairly” in our conclusion. Rather, we now point to our experiments in Figure 4 where we show that our method is more robust to noise in the data-generating process as compared to the baseline.

Action: We improved our wording in the conclusion to be more precise (see the conclusion revised Section 5).

Response to (2) discussion of failure modes

Thanks for this helpful suggestion. We added a new discussion around possible failure modes (see new Supplement A). Therein, we especially focus on potential weaknesses (e.g., explainability, scalability, etc.) that are relevant to medical practice.

Action: We added a new discussion on the applicability in medical practice (see new Supplement A).

Response to (3) abstract

Thank you. We fixed our statement. You are correct: we were referring to uncertainty-aware, neural methods for time-varying treatment effect estimation.

Action: We clarified the abstract by adding the term “neural”.

Response to (4) confounding

Thank you for your suggestion. The key motivation of our work was to highlight the effectiveness of combining neural CDEs and neural SDEs for uncertainty-aware prediction of the potential outcomes.

To address your point, we added an extension of our method where we used balanced representations to deal with confounding. Thereby, we are consistent with the baseline (i.e., TE-CDE [1]), which also uses balanced representation. Our results demonstrate that our extended method outperforms the baseline with balanced representations.

Action: We added an extension of our method with balanced representations (see new Supplement M). We find that our proposed method outperforms the baseline with balanced representations on a new real-world and a new semi-synthetic data set.

[1] 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.

[2] Yanbo Xu, Yanxun Xu, and Suchi Saria. A non-parametric Bayesian approach for estimating treatment-response curves from sparse time series. In ML4H, 2016

[3] Peter Schulam and Suchi Saria. Reliable decision support using counterfactual models. In NeurIPS, 2017.

[4] Hossein Soleimani, Adarsh Subbaswamy, and Suchi Saria. Treatment-response models for counterfactual reasoning with continuous-time, continuous-valued interventions. In UAI, 2017.

[5] Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In ICML, 2016.

Response to (1b) ODE over treatment, covariates, outcome:

Thank you for this question. Upon reading your question, we realize that we should have put more emphasis on the distinction between neural ODEs and neural CDEs and why the neural ODEs are ill-suited for our task. We are happy to provide clarification on this issue.

Why did we opt for neural CDEs instead of neural ODEs? Neural ODEs can be thought of as residual neural networks with infinitely many, infinitesimal small hidden layer transformations. Thus, neural ODEs are designed to describe how a system evolves, given its initial conditions (e.g., a patient’s initial health condition, an initial treatment decision). However, it is impossible to adjust these dynamics over time, once the neural ODE is learned. This is different from neural CDEs, which are designed to adjust for sequentially incoming information [5,6].

Why is a single neural ODE not suitable for our task? There are clear theoretical reasons why using a single neural ODE to model the dynamics of $(Y_t, X_t, A_t)$ is not suitable. If we used a neural ODE to directly model the dynamics of $(Y_t, X_t, A_t)$, this would mean that we learn an ODE that describes the evolution of $(Y_t, X_t, A_t)$, given the initial conditions $(Y_0, X_0, A_0)$. In particular, we would then make the following assumptions that would conflict with our task:

1. The health conditions of a patient are completely captured in her initial state $(Y_0, X_0, A_0)$. The initial state is the only variable that influences the evolution of a (neural) ODE. However, we want the patient encoding to be updated over time.
2. Given these initial conditions, the outcome and covariate evolution are deterministic. It is not possible to “update” an ODE at a future point in time to account for randomness in future observations. However, we want to account for random future events when they are measured.
3. The treatment assignment plan is deterministic and cannot be changed. That means, we would assume that sequences of treatments evolve like a deterministic process. This makes modeling of arbitrary future sequences of treatments impossible. In particular, using a neural ODE for the decoder of our BNCDE would mean that we can only try to predict the outcome in the future for a single, static treatment, captured in the initial state of the ODE. However, we want to model arbitrary future sequences of treatments.
4. Future observations cannot change the dynamics of the system and, importantly, cannot update our beliefs on the outcomes. However, we aim to update our beliefs on the outcome when patient information is recorded at later points in time.

How can our architecture based on neural CDEs address the above issues? Our architecture solves all of the above issues. The reason is that neural CDEs with latent representations $Z_t$ have much larger flexibility that is beneficial for our task:

1. Neural CDEs allow for patient personalization at any point in time $t$, and do not assume that all information is captured at time $t=0$.
2. Future observations $(Y_t,X_t,A_t)$ can update the neural CDE. This is crucial, because not all patient trajectories follow the same deterministic evolution.
3. Using a neural CDE in our decoder, we can model arbitrary future sequences of multiple treatments.
4. Incoming observations after time $t=0$ can update our beliefs. This makes the neural CDEs particularly suited for the Bayesian paradigm.

In summary, using neural ODEs to directly model the dynamics of $(Y_t, X_t, A_t)$ has significantly lower modeling capacities, limits patient personalization, and imposes prohibitive assumptions on the real world. To this end, we argue that it is difficult – if not impossible – to adequately model the input through a joint neural ODE in our task, and, as a remedy, we opted for an approach based on neural CDE instead.

Action: We add a new discussion in Supplement E on why neural ODEs are ill-suited for the time-varying treatment regime.

Thank you very much for your constructive feedback on our paper. We appreciate that you find the topic and the analysis in our paper interesting.

Response to (1a) latent representations:

Thank you for your question. We realized that our paper was lacking motivation for the use of latent representations. Therefore, we are happy to give more insight on this topic.

Latent representations are widely used in neural architectures, and, in particular, in the literature for time-varying treatment effect estimation. Within the literature on time-varying treatment effect estimation, there are many prominent examples: [1] and [2] use latent representations in recurrent neural network architectures, [3] in the transformer architecture, and [4] and [5] in their neural controlled differential equations. In all of the previous works, latent representations are crucial to capture the time-varying dynamics between treatments, outcomes, and covariations, which would otherwise be difficult to model without latent representations.

There are several benefits from using latent representations in our task. Latent representations allow us to capture (i) complex nonlinearities, model (ii) dependencies between observables, and (iii) reduce dimensionality in the observed patient history. For example, in our new experiments in Supplement M on real-world and semi-synthetic data, patient covariates are very high dimensional. Here, dimensionality reduction is important to encode only relevant information.

Hence, in our method, this means that $Z_t$ are used to complex patterns in and between observed outcome variables $Y_t$, covariates $X_t$, and administered treatments $A_t$. As such, latent representations are essential to capture complex interactions in the data-generating mechanism in our task.

Neural CDEs use latent representations by design. We provide a detailed discussion on neural CDEs below and in a new Supplement E.

Action: We revised our paper to clearly spell out the motivation for using latent representations.