Causal Contrastive Learning for Counterfactual Regression Over Time
Authors: Mouad El Bouchattaoui
Myriam Tami
BENOIT LEPETIT
Paul-Henry Cournède
Venue: NeurIPS 2024 poster (2024)
Keywords: Counterfactual Regression
Longitudinal Data
Contrastive Learning
Primary area: causal_inference
TL;DR: We study the problem of counterfactual regression over large time horizons leveraging contrastive learning.

Abstract: Estimating treatment effects over time holds significance in various domains, including precision medicine, epidemiology, economy, and marketing. This paper introduces a unique approach to counterfactual regression over time, emphasizing long-term predictions. Distinguishing itself from existing models like Causal Transformer, our approach highlights the efficacy of employing RNNs for long-term forecasting, complemented by Contrastive Predictive Coding (CPC) and Information Maximization (InfoMax). Emphasizing efficiency, we avoid the need for computationally expensive transformers. Leveraging CPC, our method captures long-term dependencies within time-varying confounders. Notably, recent models have disregarded the importance of invertible representation, compromising identification assumptions. To remedy this, we employ the InfoMax principle, maximizing a lower bound of mutual information between sequence data and its representation. Our method achieves state-of-the-art counterfactual estimation results using both synthetic and real-world data, marking the pioneering incorporation of Contrastive Predictive Encoding in causal inference.

Causal Contrastive Learning for Counterfactual Regression Over Time
Mouad El Bouchattaoui +1. Myriam Tamil, Benoit Lepetit", and Paul- Paul-Henry Cournede
Paris-Saclay University, CentraleSupélec, MICS Lab, Gif-sur- Gif-sur-Yvette, France Saint-Gobain, Paris, France
Abstract
Estimating treatment effects over time holds significance in various domains, in- cluding precision medicine, epidemiology, economy, and marketing. This paper introduces a unique approach to counterfactual regression over time, emphasizing long-term predictions. Distinguishing itself from existing models like Causal Trans- former, our approach highlights the efficacy of employing RNNs for long-term forecasting, complemented by Contrastive Predictive Coding (CPC) and Infor- mation Maximization (InfoMax). Emphasizing efficiency, avoid the need for computationally expensive transformers. Leveraging CPC, our method captures long-term dependencies in the presence of time-varying confounders. Notably, recent models have disregarded the importance of invertible representation, com- promising identification assumptions. To remedy this, employ the InfoMax principle, maximizing a lower bound of mutual information between sequence data and its representation. Our method achieves state-of-the-art counterfactual estimation results using both synthetic and real-world data, marking the pioneering incorporation of Contrastive Predictive Encoding in causal inference.
Introduction
It's vital in real-world applications estimate potential responses, i.e., responses under hypothetical treatment strategies. Individuals show diverse responses to the same treatment, emphasizing the need quantify individual response trajectories. This enables personalized interventions, enhancing decision making efficacy. In medical contexts, precise response estimation enables tailored treat- ments for patients (2] 70 49). This paper focuses counterfactual regression time, estimating responses under hypothetical treatment plans based on individual records, including past covariates, responses, and treatment sequences up the current prediction time 64 62 In addressing the chal- lenges of this time-varying setting, we tackle: (1) Time-dependent confounding [55]: confounders influenced by past treatment, impacting subsequent treatments and responses. (2) Selection bias: im- balanced covariate distributions across treatment regimes in observational data, requiring time-aware handling beyond methods in static settings (64| 67 (3) Long-term dependencies: enduring interdependencies among covariates, treatments, and responses, enabling long-range interactions 54].
decision-making
15
Recent advancements in neural networks, such as Recurrent Marginal Structural Networks (RMSNs) Counterfactual Recurrent Networks (CRN) (7). and G-Net (39), have tackled these causal inference challenges. However, their reliance on RNNs limits their ability to capture long-term dependencies. Recent studies 488 propose integrating transformers to better represent temporal dynamics. Rather than viewing this as a limitation of RNNs, we see it as an opportunity to em- phasize their strengths. We design specific architectures for counterfactual regression over large horizons, avoiding complex, hard-to-interpret models like transformers. Our approach leverages
42
mouadd doouadee--buccateeeee www.buacatttttuueeereeeeeeppppe
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

the computational efficiency of RNNs, incorporating Contrastive Predictive Coding (CPC) [2 29 for learning data history representations. This enhances model performance while maintaining effi- ciency, offering a compelling alternative to transformer-baset approaches. Furthermore, we usually formulate identification assumptions of counterfactual responses over the original process history space (Appendix [B.T). However, these assumptions may not hold in the representation space for arbitrary functions. Since identification often involves conditional independence, it applies when using an invertible representation function. Current models for time-varying settings [42] 7 48 do not enforce representation invertibility. To address this, instead of adding complexity with decoder, implicitly push the history process to be "reconstructable" from the encoded representation by maximizing Mutual Information (MI) between representation and input, following the InfoMax principle 43), akin to Deep InfoMax (32]
2 Contributions
Our approach is inspired by self-superrvsed learning using MI objectives [32. We aim to maximize MI between different views of the same input, introducing counterfactual regression over time through CPC to capture long-term dependencies. Additionally, we propose tractable lower bound to the original InfoMax objectives for more efficient representations. This is challenging due to the sequential nature and high dimensionality, marking novelty. We demonstrate the importance of regularization terms via an ablation study. Previous work leveraging contrastive learning for causal inference applies only to the static setting with no theoretical grounding [66]. To our knowledge, we frame for the first time the representation balancing problem from an information-theoret perspective and show that the suggested adversarial game (Theorem 5.4) yields theoretically balanced representations using the Contrastive Log-ratio Upper Bound (CLUB) of MI, computed efficiently. Key innovations of our Causal CPC model include: (1) We showcase the capability of leveraging CPC to capture long-term dependencies in the process history using InfoNCE [25] 26 (52), an unexplored area in counterfactual regression over time where its integration into process history modeling is not straightforward in causality. (2) We enforce input reconstruction from representation by contrasting the representation with its input. Such quality is generally overlooked in baselines, yet it ensures that confounding information is retained, preventing biased counterfactual estimation. (3) Applying InfoMax to process history while respecting its dynamic nature challenging. We provide a tractable lower bound to the original InfoMax problem, also bringing theoretical insights on the bound's tightness. (4) We suggest minimizing an upper bound on MI between representation and treatment to make the representation non non-predie of the treatment, using the CLUB of MI [3]. This novel cormattiontthoored perspective results a theoretically balanced representation across all treatment regimes. (5) By using a simple Gated Recurrent Unit (GRU) layer [4] as the model backbone, we demonstrate that well-designed regularizations can outperform more complex models like transformers. Finally, our experiments synthetic data (cancer simulation (24) and semi-syue cemi-synthetic data based on real-world datasets (MIMICIII 335 show the superiority of Causal CPC at accurately estimating counterfactual responses.
Related Work
Models for Counterfactual Regression Through Time Traditionally, causal inference addresses time-varying confounders using Marginal Structural Models (MSMs) [64), which rely on inverse probability of treatment weighting [62]. However, MSMs can yield high variance estimates, especially with extreme values, and are limited to pooled logistic regression, impractical for high-dimensional, dynamic data. RMSNs 41 enhance MSMs by integrating RNNs for propensity and outcome modeling. CRN (7) employs adversarial domain training with gradient reversal layer (3 to establish a treatment-invarian representation space, reducing bias induced by time-varying confounders. Similarly, G-Net (39] combines g-computation and RNNs to predict counterfactuale in dynamic treatment regimes. Causal Transformer (CT) 48 uses transformers to estimate counterfactuals over time and handles selection bias by learning a treatment- invvriiatt representation via Counterfactual Domain Confusion loss (CDC) 78). These models, like ours, assume sequential ignorability [62] in contrast to body of work which does not fully verify our assumptions 455 73
68
60,
80, 79 g 28
12 38
82 22), which we discuss in detail in Appendix C.1.2 In contrast, we introduce a contrastive learning approach to capture long-term dependencies while maintaining a simple model and ensuring high computational efficiency in both training and prediction. This demonstrates that simple models with well-designed regularization terms can still achieve high
58
69 19
31

---

prediction quality. Additionally, previous works 662 invertible representation improving counterfactual regression. Here, we introduce an InfoMax regularization term to make our encoder easier to invert. Appendix C.1 provides detailed overview of counterfactual regression models.
64
41
39
48] did not consider the role of
InfoMax Principle The InfoMax principle aims to learn representation that maximizes MI with its input [43] (]. Estimating MI for high-dimensional data is challenging, often addressed by maximizing a simple and tractable lower bound on MI 32
56 40]. Another approach involves maximizing MI between two lower-dimensional representations of different views of the same input process history into two views, past and future, and maximizing a tractable lower bound on MI between them. This encourages "reconstructable" representation of the process history. To our knowledge, the only work applying an InfoMax approach to counterfactual regression, albeit in static settings, is [16]. They propose maximizing MI between an individual's 's representation and global representation. aggregating information from all individuals into single vector. However, the global representation lacks clarity and interpretability, raising uncertainties about its theoretical underpinnings capturing confounders. Furthermore, there's lack of theoretical analysis on how minimizing MI between individual and treatment- seaument-rredictvee representations could yield a treatment-invariant representation. As novelty, we extend the InfoMax principle to longitudinal data, providing theoretical guarantee of learning balanced representations. Appendix C.2 discusses self-supervision and MI, with all proofs in Appendix G
B 29 [75 777, offering a more practical solution. We adopt this strategy by dividing
4 Problem Formulation
Setup In the framework of Potential Outcomes (PO) 655 and following (63, we track a cohort of individuals (units) i E (1,2,.... . over tma time steps. At each time t E (1,2.... mmar), J. we observe the following: (1) Discrete treat- ment Wi E W- {0, 1,..,, K - }). e.g., in medical contexts, Wa may represent treatments like radiotherapy or chemother- apy. (2) Outcome of interest Yú, E y C R, such tumor volume. (3) Time-varying context X E X C IRd,, con- taining information about the individual that may influence treatment decisions and outcomes. X is a dg-dimensional vector of confounders, such as health records or clinical mea- surements. (4) Static confounders V E V C Rd,,, such as gender, which remain constant over time. (5) Partially observed potential outcomes Y, (c), representing the outcomes that would have been observed for individual i at time t under treat- ment sequence Wi (wi,1.-..5 wi,x) W We define the history process up time + 1 as H = [V, X Wze, Yze), capturing all information prior to the assignment of treatment W++1- This history is illustrated the causal graph shown Figure I
Figure 1: Causal graph over H,+1
Goal Given a training dataset (H,,+ I.....NJ N) sampled from the empirical distribution PH.4 we address the following causal inference problem: Given a history process H how can we efficiently estimate counterfactual responses up to time t + 7 (where T > 1 is the prediction horizon) for a potential treatment sequence W44 1:147 = (We+ (+++))? The goal is to estimate the causal quantity:
H++1), i.e., the expected outcome at time L. + т, given the history HA+1 and a sequence of treatments wt+1:t+-- We identify this causal quantity from observational data using the assumption of sequential ignorability [62 42 39 48), which is implicitly assumed in Figure [ and explicitly discussed in Appendix [B.1 This allows us to express the counterfactual as:
E(Y++T (W6414477777
H = EY H441, W411477 = we+1:tt).
5 Causal CPC
5.1
Representation Learning
Contrastive Predictive Coding We employ contrastive learning to efficiently represent the process history H., Causal forecasting over multiple time horizons requires representations that capture variability in He. For short-term predictions, local features and smooth signal variations are critical,
3

---

while long-term predictions rely on capturing global structures and long-term dependencies, as shared information between history and future points diminishes.
To achieve this, we learn a representation of He that predicts future components over multi- ple time steps. For each horizon j =1,,--.. 1,....TT. T, future components are defined as Ut+j
[V, X, ttj ( EM C R(d,-d,+K+1] First, local features are extracted by encoding XV,,,,
Y--] into Z: = =, ([V, X,, ,,-1, Yt-1))- Then, the full process history H, is sum- marized into context representation Ct. given by an autoregressive model: 88 (gr(Zg) (Zgo) C. where 68 is implemented with a GRU [14]. This results in the representation function: $0 (H,) C.. To train the model, we use contrastive loss that encourages the context C, to predict future local features Z(+1.-.. Ze+T while distinguishing them from the features of other individuals. This is done by minimizing the InfoNCE loss 2 (InfoNCE))
for each horizon j:
(,, [ (0 -EB log U C.)
EIS (U
where batch containing individual histories, and T3 weight matrix. The discriminator (..)) classifies the correct future feature among negative samples from other individuals:
Ty(Ues Po, 0a (H.)) = 4, (Ue=)"" Pyc.
In practice, C, predicts Z,+j, and prediction quality is measured by the dot product 2 Zn+g-
Minimizing the InfoNCE loss c) (InfoNCE) provides a lower bound on the MI between the context
and future features l(U++j; C₁)
152
T(U [+j C+) 2 log( (B)) c. C.MSONCE).
For multiple forecasting horizons j = 1,2,....,. T, we learn long-term dependencies by minimizing the InfoNCE loss across all horizons:
(C,, 02 + C...IONCE) (0,,0,,,
Thus, minimizing CCPC maximizes the shared information between the context and future compo- nents as shown in Eq. 55 pushing the model to capture the global structure of the process over large corizans-crucial for counterfactual regression across multiple time steps:
$ Ej=1 I I(U,+,, C.) 2 log((B)) CCPC,
(5)
InfoMax Principle We introduce a regularization term to make the context representation of the process history H, "reconstructable." We leverage the InfoMax principle to maximize the MI between H, and the context C, However, we avoid computing the contrastive loss between C, and H, for two main reasons. First, H, is a high-dimensional sequence, making the loss computation very demanding, Secondly, we are still interested in incorporating inductive bias toward capturing global dependencies, this time by pushing any subsequence to be predictive of any future subsequence within H₃ Hence, divide the process history into two non-overlapping views, H = U1:40-
H. Ut +1t representing a historical subsequence and a future subsequence within the process history H,, with tn randomly chosen per batch. We then maximize the MI between the representations of these views, C and C/ resulting in lower bound to the InfoMax objective as formulated below:
Proposition 5.1. I(C,, c/) < (H,, (C), c()).
We provide an intuitive discussion of the inequality by providing an exact writing of the gap in 5.1
Theorem 5.2.
(C/ ChI) I(C], ct) = -[H, CC) En. Bep-ot of [Dizleopu [Paeee]]]
(6)
Both terms the RHS of Eq. (6) are positive, providing an alternative proof to Proposition 5.1 When equality holds, it implies I (H;; C/ C+) O, indicating that H, independent of Of given C/. This suggests that C/ retains sufficient information from H, that predictive of C/ The symmetry of MI also leads the occurrence of the second term the RHS when conditioning on Cf. The equality in Proposition 5.1 implies Pe Pc of suggesting that C/ efficiently encodes its subsequence while sharing maximum information with C..

---

By considering the proposed variant of the InfoMax principle, we can compute a contrastive bound to [(C), C more efficiently, the random vectors reside in a low-dimensional space thanks to the encoding. We define a contrastive loss using InfoNCE similar to Eq.
Clnffoaan) (0100..1) = -Bg log
(01,
expl (C, C. - (xp ,, (C C
We use a non non-linear discriminator T, parametrized by n (detailed Appendix D. The representation of the past subsequence C is mapped to prediction of the future subsequence C = Fy (C,) and T, = Cf C
Theorem 5.2 and Proposition 5.
justify using the loss in Eq. 7 by showing that our InfoMax simplification provides valid lower bound. Thus, the contrastive loss in Eq. (7) is valid, as:
log( - _ C(InfoMan) < I(C,, C S (HH, (C), C().
(8)
The "mental model" behind our regularization term comes from the MI, (H,, (C,, C()) HH,)) (H, (C), C()), which can be written using entropy. Since the entropy term is constant and parameter-free, minimizing the conditional entropy (H, (C) c 2 0 ensures that H, is almost surely function of (C), C() (Appendix G.4 Proposition G.2). When MI is maximized, the theoretical existence of such a function suggests that the learned context C, can decode and reconstruct H..
Beyond the idea of reconstruction, it was shown that the InfoNCE objective implicitly learns invert the data's 's generative model under mild assumptions [86). Recent works [18, (44) extend this insight to multi- mottil settings, which can reframe our InfoMax problem: H and H can be seen as two coupled modalities, allowing us to identify latent generative factors up to some mild indeterminacies (e.g rotations, affine mappings). We plan to extend multi- causal representation learning to the longitudinal setting, where we anticipate minimizing our InfoMax objective, in the limit of infinite data, will effectively invert the data generation process up to class of indeterminacies that we conjecture to be broader and under weaker assumptions than those in current causal representation learning literature, given focus is on causal inference rather than the identification of causal latent variables. We initiate a formal basis for this claim Appendix G.5
5.2 Balanced representation learning
Motivation Our goal is counterfactual regression, specifically estimating IEY (w1+1a+7) Hz+1). For simplicity, with т = 1, we estimate the potential outcome for given treatment W++1 = w1+1, where We+1 E {0, I.....KK K - expressed as E(Y(+1(41+1) H,-1), which under standard assumptions is identified as:
I Hz41) 1E(Y+1 He+1, W441 Wt+1).
The RHS can be estimated from data B(Y++1 He+1, W(+1) f(H W4+1). Since only one treatment is observed per individual at each time step, Wi,411 = Wi,41, our model f gener- (atee41,((4)) where w/+1 & W++1 (c.g.. chemotherapy vs. radiotherapy). The challenge is that H,+1 and We+1 are not independent, introduc- ing potential bias in counterfactual estimation [61], leading to covariate shift or selection bias. To address this, learn representation & (HH,,.) that enforces distributional balance during decoding.
ates counterfactual responses by switching treatments
Setup To mitigate selection bias, we leverage the context representation C, of H2 and introduce two sub-networks: one for response prediction and one for treatment prediction, both using a mapping of the context representation:
+ = SEL ((LneaLr(Cr))) (Ct)) = Dep. (H:),
where SELU represents the Scaled Exponential Linear Unit 37 and R denotes all parameters of the
representation learner, i.e., 0₂ [8, 8z). Following 474 our objective is to learn a representation that accurately predicts outcomes while remaining distributionally balanced across all possible
treatment choices W, 0, 1...., K - 1. To achieve this, we frame the problem as an adversarial game: one network learns to predict the next treatment from the representation, while regularization term ensures that the representation is prediceci of the treatment.
5

---

Zoom
CPC
00200
66
GRU
GRU
A
Ag
A2
GXU
GRU
GRU
W
Proceet Hirory
Historcal
Future ahsequance
Legend
Paslicdon Feaclas
pudleke Fracias
Figure 2: Causal CPC architecture: The left shows the encoder, which learns context C, from process history H,, with CPC and InfoMax objectives used for pretraining. The right shows the decoder, which autoregressively predicts the future outcome sequence from C..
Factual response prediction Since we intend to predict counterfactual responses for t steps ahead in time, we train a decoder to predict the factual responses Y++1--.., Yi+, given the sequence of treatments (We+1...., Wc+r). We minimize the negative conditional likelihood
Ly 0y) = - log Pey (yt+1:477
D2 w+418+7)
Σ
log pay (y445 Me41itj-1 Pt we4
We denote I2- [Y 1245-11 P2 W 1a41 and assume Gaussian distribution for the conditional responses Yi+/ I ((GG where Gy models the mean of the conditional response (see right side of Figure 2 We set 0.05 throughout experiments. The response sequence is estimated autoregressively using a GRU-based decoder without teacher forcing [8] to ensure model training's consistency with testing in real-world scenarios (Figure and Algorithm in Appendix [H].
Treatment prediction We learn a treatment prediction sub-netwon network parameterized by Ow that takes as input the representation P and predicts a distribution qow (W++1 ( over the treatment
W+11 by minimizing the negative log- ligeliiel,oodd, Lw log qow (We+1 Pt+1). To assess the quality of the representation in predicting the treatment, the gradient from Lw only updates the
treatment network parameters Ow and is not backpropagated through the response of the parameters for the representation 1.+1 (Algorithm 2 Appendix
Adversarial learning To create adversarial game, update the representation learning pa- rameters, and in the next step, the treatment network qaw (: @ with adverse losses such that the representation 4 becomes invariant with respect the assignment of W Different from SOTA models (as highlighted in related work) and in line with our information guidelines principles, learning balanced representation 1.41 amounts ensuring 4 .ll W which is equivalent to I [(P1+1 W++1) O. Hence, we minimize the MI as way to confuse the treatment classifier. Specifically, we minimize an upper bound on [ W141). namely the CLUB of MI (13).
TCLUB (b (H,), We+1; (
(r We41) [log 90w (We+1 ((H++)))]
- -Ep -E Br [log 90w (W++1 ((H,41))]
We use the objective in Eq. (9) to update the representation learner (.)
This update aims minimize the discrepancy between the conditional likelihood of treatments for units sampled from (H..)) and the conditional likelihood of treatments under the assumption of independent sampling from the product of marginals FH... Iw,.,1 In practice, we generate samples from the product of marginals by shuffling the treatment W++1 across the batch dimension similar to 9 32
9 32
When minimizing Lw, 4ow (Wi+1 + gets closer to the true conditional distribution p(wx+1 1, and, in this case, the objective Eq. (9) provides an upper bound of the MI between
representation and treatment. We formalize the intuition by adapting the result of [33:

---

Theorem 5.3. [13] Let 40w (D141, W(+1) = 90w (W++1 (p+1)p ( be the joint distribution induced by qow (W++1 (t+1) over the representation space of P1+1. If:
(en (#41) S D (xl(p( (ot.:ppp)n)lloeeilll (#44,
then I(1 W+11 S (CLUB(#HH) Wo+1ig).
Based on Theorem 5.3 our adversarial training is interpretable and can be explained as follows: the treatment classifier seeks minimize Ea w4+1) [Cw]. which equivalent to minimizing
Kullback-Leibler divergence Dxr (p( Pr+ 1. ) /oow ( (,.)))) Therefore, gaw (Pr+ Wh++) could get closer
we train the network to predict Wi+1 from P1+1. In such case and by Theorem 5.3 ICLUB provides an upper bound on the MI. Hence, subsequent step, we minimize ICLUB B w.r.t the representation parameters, minimizing the MI I ((11 W++1) and achieving balance. We theoretically formulate such behavior by proving in the following theorem that, the Nash equilibrium of this adversarial game, the representation exactly balanced across the different treatment regimes provided by Wi+1.
than, ultimately, to pl o ++1 (41)p(((41))
Theorem 5.4. Let t (1,2,..., & and treatment network. Let I2 (HH) be the probability distribution over the representation space and P Pa(H,)]Wea its conditional counterpart. Then, there exist 4 and q" such that:
Don and q = Qow are, respectively, any representation
4' = arg min LCwS ( (H)), W1+144)
@ qº' arg max E7 7 (H)) [log logg(Wi++) & (H,)))
(10)
Such an equilibrium holds if and only If Po.H) a. P.M almost surely.
Since target multi-timestep forecasting, covariate balancing in the representation space extends beyond / 1+1 1. For simplicity, we presented it for / 1, but in practice, the adversarial game applies the balancing across all forecasting horizons (Algorithm 2). The theorem also holds for other horizons by replacing (H,) with 4 (H++)-1) and W4+1 with W for 2 S S
Causal CPC Training The Causal CPC model is trained in two stages: (1) Encoder pretraining:
We first learn an efficient representation of the process history by minimizing loss:
Lene ==0==cCP LCPO (01, 0z. (T3))1)) + g(lnfoMue) (01, 7)-
(2) Decoder training: After pretraining, we fine-tune the encoder by optimizing the factual outcome and treatment networks the adversarial game from Theorem 5.4 Formally:
min L0 0y, Ow) (r 0y] Leum Soo (H,), Wow
Un.yy min Cw (Ow 0n) Ee,,, llug yo (W141 (Do,, (H.))-
6 Experiments
We compare Causal CPC with SOTA baselines: MSMs (64), RMSN 41). CRN 7. G-Net [39]. and
CT 48 All models are fine-tuned via grid search over hyperparameters, including architecture and optimizers. Model selection is based on mean squared error (MSE) on factual outcomes from validation set, and the same criterion is used for early stopping. Further details on hyperparameters and training procedures are provided in Appendices and
6.1 Experiments with Synthetic Data
(PK-PD) model 24 to simulate We evaluate our approach on simulated counterfactual trajectories, varying the confounding level via the parameter 7 (Appendix E. Unlike [48), who used larger datasets (10,000 for training, 1,000 for testing), we use a smaller, more challenging dataset (1,000 for training. 500 for testing) to reflect real-world data limitations. For long-horizon forecasting, we set the prediction horizon training sequence lengths: 60 and 40, with covariates of dimension 4.
Tumor Growth
We use the BinemaacoKineti--Paaaaaeeee
responses of non-small cell lung cancer patients, following previous works
and evaluate two
7

---

12-
15
04
Tler 3k4p
Thr Stop
Zime Skop
(a) 7 1. sequence length 60
(b) 7 2, sequence length 60
sequence length 60
Creak CiC
JUCH
33335
1.0
1.00
12
1.30
Tuu Skgy
Tlu Skgp
(d) 7 sequence length
(c) 7 sequence length
(1) 7 sequence length 40
Figure 3: Evolution of error (NRMSE) in estimating counterfactual responses for cancer simulation
data. Top: training sequence length 60. Bottom: training sequence length 40. In both cases, T = 10.
MSM is excluded due to high prediction errors.
Results We tested all models on the cancer simulation data across three confounding lev- els, 7 2, Figures 3a 3bb and [c] show the evolution Normalized Root Mean Squared Error (NRMSE) over counterfactual tumor volume as the prediction horizon in- creases. Causal CPC consistently outperforms all baselines at larger horizons, demonstrat- ing its effectiveness in long-term predictions. This confirms the quality of C, in pre- dicting future components across multiple time steps, capturing the global structure the process discussed in Eq. (5) Extended results provided in Appendix E.2.
In the more challenging case where the maximum se- quence length is 40 (Figures 3d
Be and
3f the етror
Cseand ctret
evolution remains similar to Figure 3a
3b and 3c Our
model maintains its advantage, outperforming most base- lines in long-term forecasting. However, Causal CPC does outperform other models in short-term forecasting, consistent limitation across experiments. Still, the model's 's superior long-term performance highlights its potential in applications requiring long-term accuracy. At higher lev- els of confounding, the model does not always outperform SOTA models at certain time steps. This may be due to the low dimensionality of the ameevaryingg components and static covariates, U, = V X, W,-1. -- which have only four dimensions. Our model leverages contrastive seaninngbassed regularization to excel on datasets with higher confounding dimensions, as demonstrated on MIMIC-III where U, has 72 dimensions. In this setting, our model consistently outperforms baselines longer prediction horizons. The occasional underperformance of Causal CPC at the final horizon is due to 10 being the last contrasted horizon, not an issue specific to T = 10. To support this, we reran all models with a sequence length of 60, T = 15, and 7 = As shown in Figure 4 Causal CPC still outperforms SOTA for horizons beyond encoder's retraining, where the InfoNCE loss is computed over all 15 time steps. The last prediction remains close to SOTA, suggesting that training over larger horizons than initially intended may be beneficial.
Time Stop
Figure Models' performance for can- cer simulation, 7 = 2, = 15.
10 due to the
6.2 Experiments with semi-synthetic and real data
Semi-synthetie MIMIC-III We used semi-synthetic dataset constructed by 48 based on the MIMIC-III dataset (35), incorporating both endogenous temporal dependencies and exoge- nous dependencies from observational patient trajectories, detailed in Appendix F.I The

---

Table I: Evolution of RMSEs for the semi-synthetic MIMIC III, sequence length 100.
Modell
tours)
10
0:00:
G-Na
0:73-0.14
+0
0.7
100
1.44 0.7
139+0
141 0.16
1.49-0
CRN
0.77 10.03
0,85100.0
0.8 4 008
0.22 0.11
0.21
0.55
0.92
0.20
1.00
0.25
1.06 1 0.28
L124 0.32
1 0.35
RMSN
0:0 0.16
000=0221
1211
0300 0.19
088 017
0.44 016
100E
015
110#014
114
L18+
patient trajectories are high-dimensional and exhibit long-range dependencies. Similar to the cancer simulation, the training data consisted of relatively few sequences (500 for training, 100 for validation, and 400 for testing). Table [ presents the mean and standard deviation of counterfactual predictions across multiple horizons (r = 10). We test two maximum se- quence lengths, 100 and 60, to assess the models' robustness for long-horizon forecasting.
Results Causal CPC consistently outperformed the base- lines, especially at larger horizons, both with sequence length of 100 and a reduced length of 60 (Figures I 5) Its superior performance at longer horizons is likely due to the high number of covariates, making it well-suited to contrastive-basee training. We also tested the models with
800/200/200 individuals for training/validationnessti in 48 (Appendix F.2.2), where Causal CPC achieved state-of-the-an (SOTA) results comparable to CT but with much shorter training and prediction times.
as
Figure 5: Performance for MIMIC III semi-synthetic, sequence length 60.
Table 2:
Models complexity and the running time av-
Table 3: Ablation study with NRMSE
eraged over five seeds. Results are reported for tu-
averaged across (1 S T S 10) for cancer
mor growth simulation (7 = 1). Hardware: GPU-
simulation (y 1) and MIMIC III.
IXNVIDIA Tesla M60.
Model
Cancer_Sim
MIMIC III
Moddd
Thrrae
tine Prodecion
Causal CPC((Pll) (Pall)
1,05
0.62
Oaani CPC (coceda deewerr)
822
l62
Cansal CPCC (wio gla//wN8))
G-Net
2.1.55
Causal CPC (wio glla/eaeeely
1.13
0.74
CRY
$2
131
41:
Causal CPC(W CDC CDCloss)
1.07
0.73
MSM
1:65
1:45
Causal CPC (wro balsacing)
1.08
0.69
Computational Efficiency and Model Complexity Efficient execution is crucial for practical deployment, especially with periodic retraining. Beyond training, challenges arise evaluating multiple counterfactual trajectories per individual, which grow exponentially with the forecasting horizon as K", where K is the number of possible treatments. This is particularly relevant when generating multiple treatment plans, such minimizing tumor volume. Table 2 shows the models' complexity (number of parameters) and running time, split between model fitting and prediction. Causal CPC is highly efficient during prediction due to its simple 1-layer GRU, similar to CRN (1-layer LSTM), while providing better ITES estimation. In contrast, CT is less efficient due to its transformer architecture and teacher forcing, which requires recursive data loading during inference. G-Net also has longer prediction times due to Monte Carlo sampling. Overall, Causal CPC strikes a strong balance between accuracy and efficiency, making well-suited under constrained resources.
Real MIMIC-III Data We evaluated our model on real MIMIC-III data, where counterfactual trajectories cannot be assessed due to the absence of observed counterfactual re- sponses. However, performance casting factual (observed) responses over time. Our model es- timates responses for each individual based on their observed treatment trajectory. As shown in Figure 6 Causal CPC consistently outperforms all baselines, especially at larger horizons, demonstrating its robustness and effectiveness in real-world settings.
still be measured by fore-
Figure 6: Evolution of RMSEs, Real MIMIC III, sequence length 100.
7 Discussion
Why does Causal CPC outperform SOTA large horizons? Our context C, is designed to capture shared information across future representations, particularly covariates, by minimizing the

---

InfoNCE loss over multiple time steps (Eq, 1. As shown in Eq. ( minimizing LCPC maximizes shared information between the context and future components, helping capture the global structure of the process. This is especially beneficial for counterfactual regression over long horizons, explaining the model's 8 superior performance. However, it may not always outperform SOTA in shorter-term predictions due to its focus on long-term dependencies.
Short-term Counterfactual Regression While our model is designed for long-term predic- tions, it may not consistently outperform SOTA for short-horizon tasks. However, the use of contrastive loss, particularly InfoNCE (Eq. suggests potential adaptability to balance both short- and long-term predictions without retraining. A trade-off could be achieved by adjust- ing the contrastive term weights across time steps Eq. ( which we leave for future work.
Ablation study We examined the model's performance in various configurations-fuu ffll model, without CPC, and without InfoMax. 3 and Figure show that remov- ing either term reduces counterfactual accuracy across all horizons, underscoring their significance. Additionally, replacing our ICLUB objective with CDC loss [48] or removing balancing increases errors. We also tested differ- ent MI lower bounds like NWJ [50] and MINE A for both CPC and InfoMax (Appendix C.2) finding that InfoNCE
Figure 7: Ablation study of Causal CPC on MIMIC III.
yielded the best results (Table 10
Full ablation results
are in Appendices E.2.2 and F.2.1
Falsifiability Test This study assumes sequential ignorability, common in causal inference 411 39 48 To assess robustness, we performed a falsifiability test by omitting certain confounders during training, while they remained in MIMIC-III data construction. As seen in Table violating sequential ignorability increased prediction errors for Causal CPC, CT, and CRN, though RMSN was less affected but underperformed at 2 2. Despite this, Causal CPC maintained its lead at larger horizons, demonstrating strong encoding of long-term dependencies.
Table 4: Results the MIMIC III when sequential ignorability is violated reported by RMSES
Model
7-3
6
Cansal CR
0.50= B:07
OXSEIDG
0.66 0.10
1007
Causal Transooneer
0.34::0.7
0.00
0.68:0006
0.06
0.00
0.09
1.00
CRN
0.40+ 0.07
0.54 0.09
0.70+ 0.09
0.84-000
0.09
0.97
0.00
100+ 013
1.18-00 0.16
0.19
1.38-021
1.30 0.23
RMSN
0.8
0.0 0.21
0.30 0.16
0.84 0.14
0.14
008+ 013
1.4 615
1094 0.18
L15=
1.200 0.23
Tightness of MI Upper Bounds Estimating MI bounds for high-dimensional variables is chal-
lenging and expensive [59] (57, often limited to low-dimensional inputs or Gaussian assumptions. In MI-constrained models, batch size is crucial. As shown in Eq. (3), increasing the batch size B tightens the lower bound via lB)( and to balance memory and performance, we chose batch sizes of 256 for the encoder and 128 for the decoder. While these bounds may not be perfectly tight, mutual information and self-supervision biases significantly enhance performance (Appendix C.2). confirmed by ablation studies. Other MI estimators like NWJ and MINE (50) 4 did not improve performance; our initial setup consistently performed better (Appendix 1.2.I).
Extending Causal CPC to Continuous Treatment Our approach could be extended continuous treatments by replacing the treatment classifier with a regressor. Since the method maximizes likelihood, the equilibrium in Theorem 5.44rmmm5. valid. However, in practice, continuous treatments will be represented by a single dimension, unlike discrete treatments with K K-dimensional one-hot encoding. This risks losing important treatment information in counterfactual predictions. A simpler adaptation to our model could involve discretizing continuous treatments.
Conclusion We introduced a novel approach to long-term counterfactual regression, combining RNNs with CPC to achieve SOTA results without relying on complex transformer models. Prioritizing computational efficiency, information (MI). Our method consistently outperforms existing models on both synthetic and real- world datasets, marking the first application of CPC in causal inference. Future work could focus on improving interpretability by integrating Shapley values into the causal framework (30]. Additionally, developing uncertainty-awir aware models tailored for longitudinal data is crucial for enhancing the
incorporated contrastive coss-based regularization guided by mutual
reliability of predictions in our causal framework 19
33 84
10

---

References
[1] Arora, H. Khandeparkar, M. Khodak, O. Plevrakis, and N. Saunshi. theoretical analysis of contrastive unsupervised representation learning. arXiv preprint arXiv: 09299999.922, 2019.
[2] O. Atan, Jordon, and M. Van der Schaar. Deep-treat: Learning optimal personalized treatments from observational data using neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
[3] P. Bachman, R. D. Hjelm, and W. Buchwalter. Learning representations by maximizing mutual information across views. Advances in neural information processing systems, 32, 2019.
[4] M. I. Belghazi, A. Baratin, Rajeshwar, S. Ozair, Y. Bengio, A. Courville, and D. Hjelm. Mutual information neural estimation. In International conference machine learning, pages 531-540. PMLR, 2018.
[5] A. J. Bell and T. J. Sejnowski. An informatioon-maximizzaiioo approach to blind separation and blind deconvolution. Neural computation, ((6):1129-1153) 1995.
[6] J. Berrevoets, A. Curth, I. Bica, E. McKinney, and M. der Schaar. Disentangled counterfactual recurrent networks for treatment effect inference over time. arXiv preprint wxXiv:2112.388// 2021.
[7] I. Bica, A. M. Alaa, J. Jordon, and M. van der Schaar. Estimating counterfactual treat- ment outcomes over time through adversarially balanced representations. arXiv preprint WxXiv:2002.04083 2020.
[8] I. Bica, A. M. Alaa, and M. der Schaar. Time series deconfounder: Estimating treatment effects over time in the presence of hidden confounders. In International Conference on Machine Learning, 2020.
[9] Brakel and Y. Bengio. Learning independent features with adversarial nets for non-linear ica. arXiv preprint arXiv: 1rX.v:7700000500 2017.
[10] D. Cao, J. Enouen, and Y. Liu. Estimating treatment effects in continuous time with hidden confounders. arXiv e-prints, pages arXiv-2302, 2023.
[11] Y. Chen, A. Prati, J. Montgomery, and R. Garnett. A multi-task gaussian process model for inferring time-varying treatment effects in panel data. In International Conference on Artificial Intelligence and Statisties, pages
PMLR, 2023.
[12] L. Cheng, R. Guo, and H. Liu. Causal mediation analysis with hidden confounders. Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining, null:null, 2021.
[13] Cheng. W. Hao, S. Dai, J. Liu, Z. Gan, and L. Carin. Club: A contrastive log-ratio upper bound of mutual information. In International conference machine learning, pages 1789-1788. PMLR, 2020.
[14] K. Cho, B. Van Merriënboer, C. Gulcehre, Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. Learning phrase representations using mnn encoder-decoder for statistical machine translation. arXiv preprint LrXiv:1406. 1078, 2014.
[15] E. Choi, Bahadori, Sun, J. Kulas, A. Schuetz, and Stewart. Retain: An interpretable predictive model for healthcare using reverse time attention mechanism. Advances in neural information processing systems, 29, 2016.
[16] Z. Chu, S. L. Rathbun, and Li. Learning infomax and domain-indepeendet representations for causal effect inference with real-world data. In Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), pages 433-441. SIAM, 2022.
[17] M. Cover. Elements of information theory. John Wiley Sons, 1999.
[18] Daunhawer, A. Bizeul, F. Palumbo, Marx, and E. Vogt. Identifiability results for multimodal contrastive learning. arXiv preprint arXiv:2303.09166 2023.
11

---

[19] E. De Brouwer, J. Gonzalez, and S. Hyland. Predicting the impact of treatments over time with uncertainty aware neural differential equations. In International Conference on Artificial Intelligence and Statistics, pages 4705-4722. PMLR, 2022.
[20] M. D. Donsker and S. S. Varadhan. Asymptotic evaluation of certain markov process expec- tations for large time, iv. Communications on pure and applied mathematics, 36(2):183-212, 1983.
[21] W. Falcon and The PyTorch Lightning team. PyTorch Lightning, Mar. 2019.
[22] D. Frauen, T. Hatt, V. Melnychuk, and S. Feuerriegel. Estimating average causal effects from patient trajectories. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pages 7586-7594, 2023.
[23] Y. Ganin and V. Lempitsky. Unsupervised domain adaptation by backpropagation. In Interna- tional conference on machine learning, pages 1180-1189. PMLR, 2015.
[24] C. Geng, H. Paganetti, and C. Grassberger. Prediction of treatment response for combined chemo-and radiation therapy for non-small cell lung cancer patients using tto-mathhmatical model. Scientific reports, 1-1):1-12, 2017.
[25] M. Gutmann and A. Hyvärinen. for unnormalized statistical models. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 297-304. JMLR Workshop and Conference Proceedings, 2010.
estimation: A new estimation principle
[26] M. U. Gutmann and A. Hyvärinen. Noise-contrastive estimation of unnormalized statistical models, with applications natural image statistics. Journal of machine learning research, 13(2), 2012. 2012.
[27] J. Z. HaoChen, C. Wei, A. Gaidon, and T. Ma. Provable guarantees for self-supervised deep learning with spectral contrastive loss. Advances in Neural Information Processing Systems, 34:5000-500 1,,,0511,22200000,,,000,,,,,,22,
[28] T. Hatt and S. Feuerriegel. Sequential deconfounding for causal inference with unobserved confounders. arXiv preprint arXiv:2104.0933 09323, 2021.
[29] O. Henaff. Data-efficient image recognition with contrastive predictive coding. In International conference on machine learning, pages 1192-419. PMLR, 2020.
[30] T. Heskes, E. Sijben, I. G. Bucur, and T. Claassen. Causal shapley values: Exploiting causal knowledge to explain individual predictions of complex models. Advances in neural information processing systems, 0144778-4889 2020.
[31] Ç. Hizh, John. A. T. Juuti, T. T. Saarinen, K. H. Pietiläinen, and P. Marttinen. Causal modeling of policy interventions from treatment-outcome sequences. International Conference on Machine Learning, pages 3050--30844 PMLR, 2023.
[32] R. D. Hjelm, A. Fedorov, S. LLvoie- Marchildon, K. Grewal, P. Bachman, A. Trischler, and Y. Bengio. Learning deep representations by mutual information estimation and maximization. In International Conference on Learning Representations, 2019.
[33] A. Jesson, S. Mindermann, U. Shalit, and Y. Gal. Identifying causal-effect inference failure with uncertainty-and models. Advances in Neural Information Processing Systems, 33: 1637- 1 11649, 2020.
[34] S. Jiang. Huang. X. Luo, and Y. Sun. Cf-gode: Continuous-timm causal inference for multi- agent dynamical systems. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD '23, page 997-1009, New York, NY, USA, 2023. Association for Computing Machinery.
[35] A. E. Johnson, T. J. Pollard, L. Shen, L.-w. H. Lehman, M. Feng, M. Ghassemi, B. Moody, Szolovits, L.. Anthony Celi, and R. G. Mark. Mimic-ili, a freely accessible critical care database. Scientific data, 3(1)11-9,001
12

---

[36] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
[37] G. Klambauer, Unterthiner, A. Mayr, and Hochreiter. Self-normalizing neural networks Advances in neural information processing systems, 30, 2017.
[38] M. Kuzmanovic, T. Hatt, and Feuerriegel. Deconfounding temporal autoencoder: estimating treatment effects over time using noisy proxies. In Machine Learning,for Health, pages 143-155, PMLR, 2021.
[39] R. Li, S. Hu, M. Lu, Y. Utsumi, Chakraborty, D. M. Sow, P. Madan, J. Li, M. F. Ghalwash, Z. Shahn, and L. wei H. Lehman. G-net: recurrent network approach g-computation for counterfactual prediction under a dynamic treatment regime. In NeurlPS 2021.
[40] P. P. Liang, Z. Deng, M. Ma, J. Zou, L.-P. Morency, and R. Salakhutdinov. Factorized contrastive learning: Going beyond multi-view redundancy. In Advances in Neural Information Processing Systems, 2023.
[41] B. Lim. Forecasting treatment responses over time using recurrent marginal structural networks advances neural information processing systems, 31, 2018.
[42] B. Lim. Forecasting treatment responses over time using recurrent marginal structural networks. advances in neural information processing systems, 31, 2018.
[43] R. Linsker. Self-organization in a perceptual network. Computer, 21(3):105-117, 1988.
[44] Y. Liu, Z. Zhang. D. Gong, B. Huang, M. Gong, A. v. Hengel, K. Zhang, and Q. Shi. Revealing multimodal contrastive representation learning through latent partial causal models. arXiv preprint arXiv:2402.06222,, 2024.
[45] M. J. Lopez and R. Gutman. Estimation of causal effects with multiple treatments: a review and new ideas. Statistical Science, pages 432 454, 2017.
[46] I. Loshchilov and F. Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.
[47] S. Matthes, Z. Han, and H. Shen. Towards unified framework of contrastive learning for disentangled representations. Advances in Neural Information Processing Systems, 36:67459- 67470, 2023.
[48] V. Melnychuk, D. Frauen, and S. Feuerriegel. Causal transformer for estimating counterfactual outcomes. ArXiv, abs/2204.07258, 2022.
[49] S. Mueller and J. Pearl. Personalized decision making conceptual introduction. Journal of Causal Inference, 11()):0220050, 2023.
[50] X. Nguyen, M. J. Wainwright, and M. I. Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847-586) 2010.
[51] S. Nowozin, B. Cseke, and R. Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. Advances in neural information processing systems, 29, 2016.
[52] A. d. Oord, Y. Li, and O. Vinyals. Representation learning with contrastive predictive coding. arXiv preprint
2018
[53] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, Köpf. E. Yang. Vitioo, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. Pytorch: An imperative style, high- petbrrrrformecc deep learning library. In Neural Information Processing Systems, 2019.
[54] Pham, T. Tran, D. Phung, and S. Venkatesh. Predicting healthcare trajectories from medical records: A deep learning approach. Journal of biomedical informatics, 69:218-229, 2017.
13

---

[55] R. W. Platt, E. F. Schisterman, and S. R. Cole. Time-modified confounding, American journal of epidemiology, 17066):687-694, 2009.
[56] B. Poole, S. Ozair, A. Van Den Oord, A. Alemi, and G. Tucker. On variational bounds of mutual information. In International Conference on Machine Learning, pages 5171-5180. PMLR, 2019.
[57] B. Poole, S. Ozair, A. Van Den Oord, A. Alemi, and G. Tucker. On variational bounds of mutual information. In International Conference on Machine Learning, pages 5171-5180, PMLR, 2019.
[58] Z. Qian, Y. Zhang, Bica, A. Wood, and M. van der Schaar. Synctwin: Treatment effect estimation with longitudinal outcomes. Advances in Neural Information Processing Systems, 84:3178-3190, 2021.
[59] T. Rainforth, A. Kosiorek, T. A. Le, C. Maddison, M. Igl, F. Wood, and Y. W. Teh. Tighter variational bounds are not necessarily better. In International Conference on Machine Leurning, pages 4277-4285. PMLR, 2018.
[60] R. Ranganath and A. Perotte. Multiple causal inference with latent confounding, arXiv preprint 1805 2018.
[61] M. Robins, Association, causation, and marginal structural models. Synthese, 121(1/2):111- 179, 1999.
[62] J. M. Robins and M. A. Hernán. Estimation of the causal effects of time-varying exposures. Longitudinal data analysis, 553:599, 2009.
[63] J. M. Robins and M. A. Hernán. Estimation of the causal effects of time-varying exposures. Longitudinal data analysis, 553:599, 2009.
[64] J. M. Robins, M. A. Heman, and B. Brumback. Marginal structural models and causal inference epidemiology, 2000.
[65] D. B. Rubin. Causal inference using potential outcomes; Design, modeling, decisions. Journal of the American Statistical Association, 100(469):322-333 2312,,33322103 2005.
[66] N. Saunshi, J. Ash, S. Goel, D. Misra, C. Zhang. S. Arora, S. Kakade, and A. Krishnamurthy. Understanding contrastive learning requires incorporating inductive biases. In International Conference on Machine Learning, pages 19250-19286. PMLR, 2022.
[67] E. F. Schisterman, S. R. Cole, and R. W. Platt. Overadjustment bias and unnecessary adjustment in epidemiologic studies. Epidemiology ( (Cambridge, Mass.), 20(4):488, 2009.
[68] P. Schulam and S. Saria. Reliable decision support using counterfactual models. Advances in neural information processing systems, 30, 2017.
[69] N. Seedat, F. Imrie, A. Bellot, Z. Qian, and M. van der Schaar. Continuous-time modeling of counterfactual outcomes using neural controlled differential equations. arXiv preprint arXiv: arXiv:2206.083//, 2022.
[70] U. Shalit. Can we learn individual-level treatment policies from clinical data? Biostatisties, 21(2):359-363 2020.
[71] R. Shwartz Ziv and Y. LeCun. To compress or compress senpress--sef--suurrviie learning and information theory: A review. Entropy, 26(3):252, 2024.
[72] H. Soleimani, J. Hensman, and S. Saria. Scalable joint models for reliable uncertainty-awar event prediction. IEEE transactions on pattern analysis and machine intelligence, 40(8):1948- 1963, 2017.
[73] H. Soleimani, A. Subbaswamy, and Saria, rreponen models for counterfactual reasoning with continuous-time continuous-valuee interventions. ArXiv, abs/1704.02038, 2017.
14

---

[74] I. Sutskever, J. Martens, G. Dahl, and G. Hinton. On the importance of initialization and momentum in deep learning. In International conference on machine learning, pages 1139- 1147. PMLR.
[75] Y. Tian, D. Krishnan, and P. Isola. Contrastive multiview coding. In Computer Vision-ECC 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part XI 16, pages 776-794, Springer, 2020.
[76] C. Tosh, A. Krishnamurthy, and D. Hsu. Contrastive learning, multi-view redundancy, and linear models. In Algorithmic Learning Theory, pages 1179-1206. PMLR, 2021.
[77] M. Tschannen, J. Djolonga, P. K. Rubenstein, S. Gelly, and M. Lucic On mutual infor- mation maximization for representation learning. In International Conference on Learning Representations, 2020.
[78) E. Tzeng, J. Hoffman, T. Darrell, and K. Saenko. Simultaneous deep transfer across domains and tasks. In Proceedings of the IEEE international conference on computer vision, pages 2015.
[79] V. Veitch, D. Sridhar, and D. Blei. Adapting text embeddings for causal inference. In Conference on Uncertainty in Artificial Intelligence, pages 919-928. PMLR, 2020.
[80] Y. Wang and D. M. Blei. The blessings of multiple causes. Journal of the American Statistical Association, 1 1154523)1574-1556 2019.
[81] R. J. Williams and Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural computation, 1 1989.
[82] S. Wu, W. Zhou, M. Chen, and S. Zhu. Counterfactual generative models for time-varying treatments. arXiv preprint arXiv:2305. 15742, 2023.
[83] Y. YazIeı, C.-S. Foo, S. Winkler, K.-H. Yap. G. Piliouras, and Chandrasckhar, The un- usual effectiveness of averaging in GAN training. In International Conference Learning Representations, 2019.
[84] M. Yin, C. Shi, Y. Wang, and D. M. Blei. Conformal sensitivity analysis for individual treatment effects. Journal of the American Statistical Association, 119(545):122-133 2024.
[85] N. Zhao, Z. Wu, W. Lau, and Lin. What makes instance discrimination good for transfer learning? arXiv preprint arXiv:0006.06606 2020.
[86] R. S. Zimmermann, Y. Sharma, S. Schneider, M. Bethge, and W. Brendel. Contrastive learning inverts the data generating process, International Conference on Machine Learning, pages 12979-12990. PMLR, 2021.
15

---

A Impact Statements
Our paper seeks to advance the field Trustworthy Machine Learning by focusing on the accurate estimation of counterfactual trajectories. This capability holds significant potential to enhance decision-making processes across various domains, particularly in healthcare, where clinicians can leverage models designed mitigate bias and promote fairness. Additionally, by focusing efficiency, our contributions extend beyond traditional machine learning considerations to address environmental concerns associated with energy consumption. By advocating for the prudent of computational resources, especially in training complex models deployed in real-world scenarios, we aim promote sustainability in developing and applying machine learning solutions.
B
Causal assumptions
B.1
Identifiability Assumptions in Causal CPC
In this section, detail the assumptions used for the identifiability of the counterfactual responses E(Y
(w++1047) Hz+1). As briefly stated in Section 3 we follows similar assumptions to
62 64 7 48
namely
Assumption B.1 (Consistency). For every time step and given any manner by which unit i receives the sequence of treatment Wi,g,, we always observe the potential outcome (ws,g)) Formally:
W Wiigt
Ya (u
Assumption B.2 (Sequential Ignorability). Given any time step t, we have the conditional indepen- dence:
(u(())) W., Ha = has M(sie hir)
Assumption B.3 (Overlap/positiiitt)) Given any time step i, and for any possible historical context h, the probability of observing any of the possible treatment regimes strictly positive but not deterministic:
pha) + 0 = < p( we (h))
The three assumptions are sufficient for the identification of the counterfactual responses from observational data, which we formulate in the following proposition.
Proposition B.4. Assuming consistency, overlap, and ignorability (assumptions B.
B.2
B.3
the
causal quantity EYE++r (W1+1at4))
H(11) is identifiable from observational data following
TE (w2+11477
H++1) (Y+++ H4+1, W41147 = w(+1:(7))
Proof. See
62
B.2 On the Causal Graph
We repeat the causal graph introduced in Figure I to explain the data generation process. here, all of the past observed data encompassed in H,+1 confounds future treatments and responses, W441, Wt+2..... Wimaz and Y4+1, Y:42..... Yemar> which create long-term dependencies. The fact that post-covariates affected by past treatments creates time-dependent confounding. The static covariates are assumed to be affecting all of the time-varying variables. Since we suppose sequential ignorability, there
possible exogenous noises affecting both treatments and responses. However, such noise may possibly affect responses, time-varying covariates, and response variables.
In the figure, for simplicity, we represent past treatments as W such that each element in that sub-sequence confounds next treatment and response W4+1 and Y,+1- Idem for Yst and X The static covariates V are assumed to be affecting all the time-varying variables. We omit the representation of exogenous noise for simplicity. Interactions berween W St X and s were also omitted for simplicity.
16

---

X++
C Extended related work
C.1
Counterfactual regression over time: Methods overview
C.1.1
Methods included in experiments
In this section, we give brief overview of models included in our experiments: MSMs (62| RMSN
41, CRN 77] G-Net (39), and CT [48). To delineate the differences between these models and Causal CPC, we detail in Table 5 the main design differences between all these models.
Table 5: summary the methods included in our experiments
Bacthone
harding
fencas?
remn dependencies
Tramiig
factials
bia
senratinn
Cauadl CPC (ours)
GRL
CPC
Icam long tenn mis
Autorgarcsivve
Balazeed repecamns
yo, contraat acoscich
Inpua
GA0 CRN
ESTW
KIA
LSTH
No
NA
NA
Autoegressive
Balixedd repusenn
NA
50u
Logantccinex
Anoregresise
Wagtiing
C.1.2 Methods Violating Our Assumptions
Our work relies on assumption of sequential ignorability, yet several alternative models operate under different assumptions, often addressing the presence of unobserved confounders. Some of these models are rooted in deconfounding theory [45] 60 80]. which has been extended to time-varying settings. Deconfounding involves imposing factor model on treatment assignment, where each treatment becomes conditionally independent given latent variables that act as proxies for unobserved confounders. Examples of this approach include 8. 28 10]. Other models assume the presence of proxy variables, inferring a representation of unobserved confounders through probabilistic models based these proxies 779 12 38).
In contrast to our setting, which is governed by the three causal assumptions in Appendix B.I many models assume aatta-ennrattin process similar to [73 72 58 These methods, often non- or semi- parametric, tend either ignore static covariates or handle them linearly, leading to computational inefficiencies and scalability issues. Nevertheless, some non- or semi-parametric approaches-sun as (68 69 19 31 -aiggn with our causal assumptions but extend them to continuous time, treating sequential ignorability in a continuous setting.
Additionally, models like 34 incorporate continuous-time and assume interactions between units, where an individual's outcome depends on both their treatment and the treatments others. [6], focus- ing binary treatment sequences, requires stronger version of sequential ignorability -onaaiiiiyy-onniioonnnll on current wwariates--wereae our model assumes weaker version, conditioning on the entire history of covariates account for long-lasting confounding effects.
Furthermore, focuses solely on binary treatments and targets the estimation of the average treat- ment effect on the treated (ATT). The authors assume specific treatment regime, where individuals enter post-treatment state after defined point in time. This assumption restrictive compared to framework, which allows for complex, individualized treatment assignment mechanisms
17

---

and non-binary treatments, where treatment values can fluctuate over time. As result, is incompatible with our causal assumptions.
Other methods, like [82], address high-dimensional counterfactual generation based on time-varying treatment plans under the same sequential ignorability assumption. However, they are not designed
for causal forecasting over multiple time steps, as required in our setting. Similarly, [22] focuses
on estimating the average causal effect and is not suited for predicting individual treatment effects or conditional counterfactual responses, as it targets marginal counterfactual expectations via computation.
C.2 Mutual Information and Self-Supervision
Self-Supervised Learning and Mutual Information In self-supervised learning, Deep InfoMax uses MI computation between input images and their representations, focusing maximizing MI to improve reconstruction quality. Local MI between representations and image patches captures detailed patterns across regions, enhancing encoding. By maximizing average MI with local regions, Deep InfoMax significantly boosts downstream task performance, while global MI plays key role in reconstructing the entire input from the representation.
32
CPC aligns with the MI-based approach seen in Deep InfoMax, emphasizing the maximization of MI between global and local representation pairs. Distinct from Deep InfoMax, CPC processes local features sequentially, constructing partial "summary features" " to predict specific local features in the future. While classical self-supervised paradigms often focus on tasks like classification or reconstruction-bat
abjectives-favoriie either local or global MI maximization--intgraalin approaches becomes essential for downstream tasks such as counterfactual regression over time, which justifies why Causal CPC is designed to support both local and global MI maximization to improve temporal predictions.
both
Several other methods share similarities with CPC, such as Contrastive Multiview Coding 75 This method emphasizes maximizing mutual information between representations of different views of the same observation. Augmented Multiscale Deep InfoMax [B) akin CPC, makes predictions across space but differs by predicting representations across layers in the model. While Instance Dis- crimination [85] encourages representations capable of discriminating between individual examples in the dataset, our preference for CPC arises from its adaptability in processing sequential features in an ordered and autoregressive manner, which aligns with the requirements of our specific context, especially when dealing with counterfactual regression over time.
Mutual Information and Inductive Bias. Mutual information (MI) estimation success relies not only on MI's properties but also on the inductive biases from feature representation choices and MI estimator parameterization 777.. Experimental evidence shows that, although MI remains invariant under homeomorphisms, maximization with an invertible encoder during random initialization enhances downstream performance. While higher-capacity crities yield tighter MI bounds, findings consistent with [59] suggest that simpler critics provide better representations, even with looser MI bounds. Accordingly, we selected simple bilinear critic function for contrastive losses. In vision tasks, augmentations and contrastive loss properties are crucial for representation efficiency 76, 27), and (66) highlights that inductive bias, via function class representation and optimizers, significantly affects downstream performance, offering theoretical, non-vacuous guarantees on representation quality.
Variational Approaches and MI Estimation Challenges The estimation of MI faces inherent challenges, particularly within variational lower bounds. These bounds often degrade as MI increases, creating a delicate trade-off between high bias and high variance. To address this, methods that utilize upper bounds on MI have been developed, attempting to mitigate challenges associated with variational bounds. One strategy for MI maximization involves computing gradients of a lower MI bound concerning the parameters of stochastic encoder. This computational approach potentially eliminates the need for direct MI estimation, providing more tractable solution. However, estimating MI from samples remains challenging, and traditional approaches encounter scalability issues in modern machine-learning problems.
It's crucial to note that higher estimated MI between observations and learned representations does not consistently translate to improved predictive performance downstream supervised learning tasks. CPC is an example, exhibiting less variance but more bias, with estimates capped at log [8]. Strategies to reduce bias, such as increasing the batch size, introduce higher computational complexity, requiring additional evaluations for estimating each batch with the encoding function.
18

---

In our empirical approach, we adopt specific sampling strategy for sequences, considering one- time step per batch. This facilitates computing the InfoNCE between local summary features at time and the future prediction of local features, leading to reduction in algorithmic complexity for contrastive loss computation. Empirical observations demonstrate non-decreased representation quality and improved prediction of factual and counterfactual outcomes.
Other MI lower bounds. The Mutual Information Neural Estimator (MINE) 4 leverages the relationship between MI and the Kullback-Leibbe (KL. divergence. MI can be expressed as the KL divergence between the joint distribution and the product of marginals:
I(X; Z) == (x IP @ Pz)
MINE employs the Donsker- Varadhan representation 20 of the KL divergence:
DK
sup T-2
Ee[T] log (Eo(e"))
(11)
Here, the supremum is taken over all functions T where the expectations exist. For specific class of functions F. potentially represented by class of neural networks, we obtain the lower bound:
DKL(PI1Q) 2 sup Eg log (Eq(e"))
(12)
TEF
In practice, maximize
MMINE = - log (Boee")))
where T, a discriminator parameterized by 7, representing neural network parameters. The MINE estimator is strongly consistent estimator of the true MI (Theorem 2, (
Alternatively, the f-divergence representation of Dxz [51 allows to derive another MI lower
bound, known as the Nguyen, Wainwright, and Jordan (NWJ) estimator [50]:
DKL
(1Q) sup Ep [T] log (Fole"-1)
(13)
TEF
This results in the estimator:
jNWJ (P, Q) = Eg [T,,]] - log (Bale"--11)
Unlike the InfoNCE estimator, which exhibits high bias and low variance, the NWJ estimator has low bias but high variance [57].
D Experimental protocol
All models were implemented using PyTorrc [53] and PyTorch Lightning 21. In contrast the approach in [48] we employed early stopping for all models. The stopping criterion was defined as the Mean Squared Error over factual outcomes for dedicated validation dataset. Specifically, for the Causal CPC encoder, the stopping criterion was determined by the validation loss of the encoder.
While all models in the benchmark were trained using the Adam optimizer [36), we opted for training Causal CPC (encoder plus decoder without the treatment subnetwork) with AdamW [46] due to its observed stability during training. Similar to the common practice in training GAN discriminators, the treatment subnetwork was optimized using SGD with momentum 77
The CT employed the Exponential Moving Average (EMA) 83 of parameters to enhance training stability. However, this technique not applied to Causal CPC, as experimental evidence suggested only marginal improvements. Weight decay was set to zero for all models.
For each experiment, the models were trained over five different seeds, and the reported performance metrics include the mean and standard deviation of the results.
The counterfactual trajectories are generated following two strategies:
Trajectories are generated with a single treatment per tra- jectory while the treatment slides over the forecasting range to generate multiple trajectories. Similar to 77 we apply such a generation scheme to cancer simulation data.
Single sliding treatment
Random trajectories: Trajectories are generated such that at each time step, treatment is generated randomly. We apply random trajectories to semi-synthetic MIMIC data.
For the falsifiability test on MIMIC III datset, we mask two confounders from the inputs of the benchmark models, namely sodium and glucose measurements.
19

---

E Experiments on synthetic data: Details
E.1 Description of the Simulation Model
We present a tumor growth simulation model, focusing on the PharmacoKinetie-PParaaaa PharmacoDynnmii (PK-PD) model as discussed in [24]. recent approach to predicting treatment responses in non-small cell lung cancer patients. This simulation models the evolution of tumor volume, denoted by Vtt), in discrete time, where / represents the number of days since diagnosis:
K V(I
V(t)
+ Aog
(tt(tt)
-(iraRd((t + vRd(t))) +
v(t- (14)
Chematherapy
Radiation
Noise
Tumor Growth
Here, the model parameters A, K Ke, Bed, U are sampled for each patient based on prior distributions from [24]. Additionally, Rd(t) represents the radiation dose applied at time and C(t) denotes the drug concentration.
We introduce confounding into the assignment of radiotherapy/chemottraayy treatment by making it dependent on past tumor volume evolution. Treatment is simulated using Bernoulli distribution with probability ( where:
(D(t)
(15)
D max
In this context: (tt) represents the average tumor diameter over the last 15 days, Dmax = 13 cm is the maximum tumor diameter, 5 set to d = D2 The parameter 7 controls the level of time-dependent confounding, with a higher y value assigning more weight to the history of tumor diameter in treatment assignment.
E.2
Additional results
E.2.1
Comparison benchmark models
We report in this section detailed counterfactual errors for Causal CPC and baselines over the cancer simulation dataset, which are responsible for Figure
Table 6: Results the synthetic data set with sequence length 60: mean standard deviation of NRMSEs. The best value for each metric is given in bold: smaller better.
Model
111
Casool
C... (onrs)
B.83T 0.06
0.86 T OJR0
BUWE 0.19
033 0.8
1ak 0.10
La7
L12T 0.00
L.7 4.09
LA2 BA6
CT
0011
108=0.15
0118
1.11
+0.11
121
4 17
0.000
1:35 0.16
G-Net
1911011
F1100.6
12610116
0.17
13010.18
14710.19
1S01018
1S711022
1.3-0:22
1:210225
CRN
100-1.11
10
L17230.14
13238018
13321223
RMSN
0994011
0.944004
13049665
145-0.76
136-033
186-038
1234001
1.74089
181-6.8
1884-086
MSM
12010010
18310.25
20710.44
238.0.44
134
13010.37
3011 18
30850035
3,05:0036
0.66-0.22
0.95
1.07
12413220
L.331015
1272021
0.26-025
1.44:0.29
1.55:0227
1.440226
1,0110200
1
1475027
13311025
13030.23
1.3-0.24
13360026
1.45:0228
1.5050311
1,001025
167-0.38
1,75:042
MSM
LA32022
2223033
20734003
298
1.9200.4
2.3020.77
3A:20770
3AR15025
1.34
7-3
Cansal (PC
L31031
L361827
12610.36
0.3
133
100 1.47
1811633
2001831
214.061
1.2810.66
CT
LAM3003
178:033
117520000
1:233000
231
1.3720173
G-Nat
134-6.24
1224018
5449026
127-0.39
1.4-0036
3.89-040
2344047
2.4-6.50
3,83+0
CRN
1361019
5511038
L301048
136.050
1981039
0.03
RMSN
2966009
13580.4
1355013
206:0001
214:0.6
E.2.2 Ablation study
We detail here the results of the ablation study conducted on the cancer simulation dataset (Table 3
The (full) Causal CPC model, as presented in the core paper, gives, in most cases, better results than any ablation configuration.
20

---

Table 7: Results on the synthetic data set with sequence length 40: staddard deviation of NRMSEs. The best value for each metric is given in bold: smaller is better.
Model CT 1.050009
L1320.12
154:022
1.30
L432011
1.52 2219
1.3910.22
025
1.110.25
76.021
1.810021
1801018
G-Net
L11=20100
130=20.17
152
1.2620.24
86:4029
1.9620.34
108:0.40
2.1320.40
2. 1992022
CRN
1.01-0.12
141:0.14
11910.14
1.30-0.15
144140.12
149-0.09
1.5610.07
162.005
16740.04
1.700.01
RMIN
1.32.008
1,000008
1.710.08
1610006
10110.05
-
Causal CPC (ours)
L412009
13020011
13320.20
141:025
1.5100288
1.560031
10010.23
166.0027
1201026
1.741025
CT
16.00.1
1.3110.9
1.010:35
1,5110.00
1.0010.31
17510031
1.2210.22
29
190
2.0220:40
112-0
5:220500
23::055
CRN
1.24.022
1211:0118
1.3110.32
10255025
1.5510.29
163-033
17910.41
2.0510..59
213310770
RMSN
12810.11
1:2810.15
1.720012
186210.19
176.025
1.8910300
20310.36
2.1510:1
2.2610.466
15210.19
0.210023
1338
15210.24
1.910.27
1.3
2.0210.38
CT
1.3610.18
18811033
1.2210.34
135 00.5
2.0510885
2.19-092
2.2911.00
2A511.09
2.4311077
G-Net
L19:2002
1.3820.04
1.72
1.92=0.08
2.11:0.188
2.32-0229
2.52:0.41
2.72:0.54
2.9320.58
3.13
CRN
1.000001
1,4710007
16210.2
1.76.0:40
18910.31
198_0057
200510.61
10.00022
2,13100.1
2.1510.38
RMIN
T.000013
20011061
214
Table 8: Results of the ablation study on the synthetic data set: mtandsttandadd deviation of Normalized Rooted Mean Squared Errors (NRMSEs). The best value for each metric is given in bold:
smaller is better.
Wedel
Canal CPC 20rEERNCE)
091360
1.10
Canal CPC (wle pragetrer
0843004
03660.09
091:0.88
0.9-0.10
10746008
1.66 0.8
0.6:0000
+3120012
13840008
146-9.10
Ceusal CPC CBC Toot
D2915000
140= 007
1J02 0D7
115
128011111
Cassal CFC (w Baloncing
GME0N
0331500
LIME
1.202
0.10
1255000
1.29:3112
F
Experiments on semi-synthetic data: Details
F1 Deseription of the simulation model
In this section, we provide a concise overview of the simulation model built upon the MIMIC III dataset, as introduced by 48 Initially, a cohort of 1,000 patients extracted from the MIMIC III
data, and the simulation proposed by [48] extends the model of [68].
Let dy be the dimension of the outcome variable. In the case of multiple outcomes, untreated
outcomes, denoted as () for j =1 =1,...,; .y. are generated for each patient within the cohort. The
generation process is defined as follows:
z B-spiint)) + + ofghh00) + a2 ((
(x+)++
+ Et
(16)
endogenous
caogenous
where: the B-spline B-spline(t) is an endogenous component, ())( )) is sampled independently for each patient from Gaussian process with a Matérn kernel and f2 (.) is sampled from Random Fourier Features (RFF) approximation of a Gaussian process.
To introduce confounding in the assignment mechanism, current time- varying covariates are incorpo- rated via random function f (X+) and the average of the subset of the previous Tj treated outcomes, År, (Y.-).. For d₂ binary treatments A!, where = 1,..., da. the assignment mechanism is mod- eled as:
PAj a (1'Ann (Y--1) + A' (X,)+ bi).
A ~ Bernoulli (PAS)-
Subsequently, treatments are applied the untreated outcomes using the following expression:
mill- d, IV [AA=11PAA88.
Σ
(17)
(wl i)²
inf-po
The final outcome combines the treatment effect and the untreated simulated outcome:
= E
(18)
21

---

F.2
Additional results
F.2.1
Ablation study
We detail here the results of the ablation study conducted on the MIMIC III semi-synthetic dataset (Table 3). The (full) Causal CPC model, presented in the core paper, gives consistently better results than any ablation configuration.
Table 9: Results on MIMIC III semi-synthetic data set: standast deviation of Normalized Rooted Mean Squared Errors (NRMSEs). The best value for each metric is given in bold: smaller is better.
Causal
Causal CPC (wo (
0.35 0:04
0.01.0.0
0.99 000
0.04 D.06
0212 008
0:554:0000
0.77
007
0.700.000
0.112 000
0.81 0:00
Cansal CPC (win (0mpstii
0.36 0.0
0.0
6.6-
010
0.77
0.71 000
0.50
005
0.00
0.08
006
Causal CPC (CDC (mo)
0.362 002
0,020000
028
005
0008
0.712
008
0.001
000
0.332
007
005100000
0.362
003
Consal CPCo balinding
0.352 003
03001
000
0.00
005
0000
0.72E
1006
0.162
0006
0.782
0066
0100
0.06
0.832 0036
(OSSE 0000
Furthermore, We replace the InfoNCE objective used to compute the CPC term and InfoMax terms with that of NWJ and MINE (Section C.2). We repeat the same MIMIC III experimentation while varying the objective used for CPC and InfoMax. Table T0 shows the counterfactual errors for each configuration compared to the original formulation of Causal CPC. In all cases, The InfoNCE objective performs better with notable error reduction at large horizons.
Table 10: Results of NWJ and MINE MI lower bounds when used for CPC and InfoMax for MIMIC III semi-synthetic data set: -tandat deviation of Normalized Rooted Mean Squared Errors (NRMSEs). The best value for each metric given bold: smaller better. Model
Origiial
0.88
0.06
CPC with NWJ
0.3L 000
0.8 0.08
0.5%_
0.06
0.06LL 0.07
0.71 0.08
0,751
0.28 0.07
0.1 0.06
0.80 0.06
0.87 0.05
CPC with SINE
D1352 800
0.302
0103
U.01= 1004
0.00E 1..4
0.75= 1.04
0.792 0.03
0.82= 0003
0.852
0.80 0.12
GVT= D.02
InfoMax with NWJ
0.42 0.08
0.56.t 0.04
0.89 0.07
0.77 0.08
0.83- 0.00
0.87 0.09
0.00- 0.09
0.92: 0.09
0.94 0.08
0.96: 0.08
TalaMax with MINE
0.9 006
033F 020
0.33 0006
0.74 83
0.00
0.10
0.11
6-8--.11 0.11
0397
0.10
091- 0.10
093- D007
F.2.2 Comparison to benchmark models: standard train/test split
As mentioned in Section 6.2 We also tested Causal CPC on MIMIC III semi-synthetic data using the same experimental protocol as 48), namely by using the split of patients into train/validation/test as 800/200/200. As result, baseline performances in Table 11 are exactly the same as in [88
are
Table 11: Results over the MIMIC III semi-synthetic data set (same experimental protocol as in 48) sstadadd deviation Rooted Mean Squared Errors (RMSEs). The best value for each metric is given in bold: smaller is better.
Madel
Cameil CPC (ours)
0:28 0.00
0.37 + 0.02
0.40
- 0.00
0.48
0.11
0.49
0,02
0.52 0
0.58
0.83
0,56 + 0.03
0.58 0.64
0.60 0,83
CT
0.81
0.35 0000
U.45 0.00
0.19
0.32
00210002 0.02
0.53
0.00
0.5
0002
0.56
356616002 0.02
0.38 0,02
0.39
0.02
6.Sat
1.34-1000
0177
013
1..33
1.14
1144
1104
1.13
0105
1:10+00000
116
1.21
6.066000 0106
1.25 07.06
1.29
1000
CRN
0.30 0.01
068
0.02
0.39
0.00
0.6
0102
0.02
0:41
0,01
0.0
0.24
0.16 0.01
0.76
17762.00 0002
RMSN
024 0.01
0.67 0.01
0.60
0.00
0.70
0.00
0.78 +. 0.04
0.84 0.05
0.89
0.6
0.94 0.00
0.97 0:00
1.00: 0.11
MSM
0.74
000
F22.3 Running time and model complexity
In this section, we complement the table about complexity and running time given for cancer simulation in the core paper by providing the exact same table but for MIMIC III semi-synthetic data.
G Proofs of theoretical results
G.1 Relation between InfoNCE loss and mutual information Proposition G.1.
1(U++,, C₁) 2 log( llg( -. (InfoNCE)
22

---

Table 12: The number of parameters train for each model after hyper papee--rraeetrrs fine-tuning and the corresponding running time averaged over five seeds. Results are reported for semi- semi-synthetic MIMIC III data; the processing unit GPU- NVIDIA Tesla M60.
TRAINARLE PARAMETERS
TRAINING TIME (MIN)
PRRDICTION TIME (MIN)
CALSAL CPC (OURS)
9.8
4=
CT
12
[4+]
18-2
G-NET
14.7
7_1
40
CRN
15.1
21#2
RMSN
20
4824
Proof. In the following, we draw inspiration from the proof of [52]. The InfoNCE loss corresponds to the categorical cross-entropy of classifying the positive sample U2+s correctly, given the context C₁ with a probability:
( C.))
Ela cxp(T, (UL,t) C.)
The positive sample Ut+ is one element in the batch B, where the remaining elements serve as negative samples. Let pos E (1....,|55 be the indicator of the positive sample Uz+j- The optimal probability is given by:
pl t
C+) IIl (B pl (uue+s)
p Ugon App...t..ll..... [+)
p[Index = pos B, C.)
Upo.tt+]
B 5=1
puu
C₂) II. uzj p(u
p(u,,r,, p(u,r+)]
For the score exp(T) C:)) to be optimal, it should be proportional to p(ppr.teyll..) (c.) The mutual information (MI) lower bound arises from the fact that (Cp(T)(U14H) C.)) estimates the density ratio pp (c))
plupe
p(Uppa,t+))
p(u..,/ec
C c(InfONCE) EB log
p(ur-1]
p(ur+y/ed) p(u(+s)
Σ EB
au [c]
p(u,)4)))
1 P (U+tz)
p (ur.e+s ct) p pw),e+))
Eg log
Pur++
Ce)
Eg log
1 + p (uie++
p (u
(IB) 1))E
(urett ct) (u
(19)
Ce)
p (uc+s)
C+) (1B) - 1)
Eg log
1 +
(ue+y
(u++j)
2 Eg log
B
[Ur+j C2)
(u++j. ct) + log((B))
The approximation in the third equation, Eq. becomes more precise as the batch size increases.
G.2 Relation between InfoMax and Input Reconstruction
We now prove Proposition 5.1 which states that:
I(C,, C s I(H, (C,,,(C,.C().....
Proof. This follows from two applications of the data processing inequality (17,, which states that for random variables A, B, and C' satisfying the Markov relation A - B + C, the inequality [(A; C*) < [AA; B) holds.
First, since C. &0,,0, (HA) and C/ oo 02 (H.).
write H truncy (H,) and
H = trunc;(H,, where truncy and trunc, truncate the future and history processes, respectively, given a splitting time to.
23

---

Thus, we have the Markov relation:
C
$ otruncy H
C
which is Markov equivalent
C 20,.02 otruncy
Day 0y otruncs C.
H
By the data processing inequality, this results in 1 (C), о - IH,, C.). ) On the other hand, we have the trivial Markov relation H; - (C, C/) + C which implies [H, C I [H,, (C, C )). Combining these inequalities proves the proposition.
G.3 Proof of Theorem 5.2
To begin, we split the process history into two non-overlapping views (Figure 2): H := U and
H + I:t representing historical subsequence and future subsequence within the process history H,, respectively. We then computed representations of these two views denoted C and C/ respectively, This naturally gives rise to the Markov chain, as in showed in the proof of proposition
5.1
C + H, - C.
which is Markov equivalent to:
H
Following this Markov chain, we can show that [71]
ICc Ct) I(H, C En Bef Ih
On the other hand, by applying the chain rule of the mutual information to I(H,; (C), c()) we get:
THH (C/, (C/, c = 7(H,, C + T(H,; c/ C
Combining these equations, the tightness of our bounds can be written as:
I(H;; (C/, - C I(H;; of c+ + Enc-PH, Red of mh
DKr. (Po: In
[
G.4 On the Relation Between Conditional Entropy and Reconstruction
We now prove the statement in the core paper, which asserts that the conditional entropy HH, (C).
from 2 is minimized if H, is almost surely a function of (C,, C.). The proof is adapted
Proposition G.2. HA B) = 0, then A f(B) almost surely.
Proof. For simplicity, suppose A and B are discrete random variables. Assume, by contradiction. that there exists by and two distinct values a and ag such that p(a: bo) and p(az bo) > Then, the conditional entropy is given by:
H(A B)
p(()) (p(b))dd p(a log b).
b
In particular, we have:
H(A B) 2 p(bo) (-p(a: bo) log bo) p(az bo) log bo)) > 0.
Since -llog! 2 for < S and strictly positive for f not equal to 0 or the conditional entropy H(A) B) 0 if and only if A is almost surely function of B.

---

G.5 On the benefit of the InfoMax loss on inverting the data generation process
To ensure identifiability in the latent space, we leverage recent advances causal and disentangled
representation learning. Suppose the true data-generating process is given by H, g(ze), where z, represents the true latent factors. In the sequential context, we assume that the same function generates two historical subsequences:
H g(((), H = g(z,).
We assume general dependency of the form:
plz 2 = ((z() ((zh)
expl exp(-d(z/_ z!))).
Here, is an encoder, and we use the InfoMax regularization term as follows:
( (H))))
((InfoMaz) (d := EE log
Σ 8 1=1 exp(-d(@(H)) )H)
According to [47]. under certain conditions, if the encoder minimizes ((InfoMax) then = o f is scaled permutation matrix. This result suggests that when the encoder achieves minimizer for ((IufoMes) the encoder function f closely approximates an invertible transformation of g. From a causal inference perspective, if Yir(w)) TL Wie Ha and Has g (Za), then invertible function g o ensures that:
(wir) TL Wie I f(Ha).
Thus, (w() IL Wa ((C))) and since g is invertible, we have:
Yu(wa) TT War Ca-
This demonstrates that the representation C₁ retains the essential independence structure, facilitating accurate counterfactual inference.
G.6 Proof of theorem 5.4
To prove the Theorem 5.4 we first prove the following lemma and proposition.
Lemma G.3. Let 4 be a fixed representation function. Given that q (W ((H))) is the conditional likelihood of observing the treatment 141, denote the probability of observing each treatment value
as q6 = q( (H.)) == q(W++1 =j 4b (H,)) for E 1, 1........K K 111} Then, the optimal treatment prediction function is such that
p (HH,) W4+1 j)
p (T(H)))
(20)
K-1 Σk- (H.) W+1 1)p(() = =))
Proof. For fixed representation C finding the optimal treatment probabilities amounts to solving the following optimization problem:
K
max (e(o [log (WW++ ((H))) subject to
(HHH)) =
(21)
First, we write the likelihood g(W++1] (HH))) using the conditional probabilities ()((H)))
qWe+1 (HH))
II o' (HH)) (WW+1=13
j=0
25

---

Then, the treatment likelihood can be written as
K-1 log( ( (wen1-))
(.( llo (WW ( Ep(o(H))Ween))
(w (W
logid (WW))))(W) =1 ( /
E
f (*(,))) W4+1 = () (H,
Let's denote or = p( = 1), the marginal probability of observing the 1-th treatment regime, and pi (H,) = (H,) W4+1 = 1) with a corresponding probability distribution P. We intend to maximize point-wise the objective in Eq. (21). Plugging the latter formulation of the conditional likelihood Eq. (2T) and writing the Lagrangian function, we get
K-
K-1
max
logid ( (He)on + X 2 ((HH)) 1)
(22)
1=0
1=0
Computing the gradient w.r.t of (H, for / E (0.1.....,4 K - 1} and setting to zero, we have
(*H,) aip) (H+)
(23)
Then, by the equality constraint, we find that À - oup) (H,).
Proposition G.A. Let & be a fixed representation funccitotoionnnniin The ICLUB objective when the treatment prediction function is optimal (i.e. q = x²) has the following form:
K-1
K-1
ICLUB Σ DDDK((P))
Σ
aPpp) + BRocn
[(HH)))
(24)
1=0
Proof. First, recall that
(crun(((HH) W W++134")= = EP (0 W441) [loga" (We+1 ((H++1))] [B]]-EE,,a E Br (loga" (We41 | ((He))))
(o Wo+li")) = A-A B
Let's detail A and B separately,
A=
0g log(a' (H,) (H.
K
ajp (H₂)
ag log(
2p (H,)
Σ pi (H, )au
K-1
p. (H,)
Σ
aj log(
1 ( + log(a,)a,
21=0 p H.)
K
J=
K-1
K-1
K-1
Σ
ay
+
log(a,)a
j=0
j=
Finally, we can write
K-1
K-1 Σ
A
Σ
a, DKL
- H((w11)
(25)
1=0
26

---

For the remaining term B, we have
B = Ep Poan Be (log (We+1 (HH++))
K-1
BP Er [log(q' ( (Wi
K-1
Σ BPo(H)))
|a, (og()((((H))))
K-1
agpj (H+)
Σ
aj
log
(
E/= Pt (H, )a
J=0
K-1
pl (())H)))
p(W++1 = ((H)))
Σ
aj
log
Σ Ei=0 Pi (H₂)as
K-1
p(W
p(((H)))d
1=0
H(W++1)
K-1
p( (H,)
Σ
aj
log
Σ K-1 1 L1=0 Pi (H,)ar
K
p( =j ((H)))
Σ
Ojj
log
(H) H(W+1)
p(W(+1=))
f=n
The final form of therefore
/ (Pw, (0.) H(W++1)
(26)
The proposition follows immediately from Equations
25 and 26
Proof. (Theorem 5.4 Since by lemma G.3 the TCLUB formulation in proposition G.4 holds, then to prove that the representation is balanced, it is enough to see that by the positivity of DEL
TCLUB 2 Broas, [DKc (P IPw.. (H..) ) 2 0
(27)
IoluR is minimal when ICLUB = 0, which happens if and only if for E (0. K - 1} ((H.)) almost surely which, by Bayes rule is equivalent to say p()(H,)) = = (H,) W441 =])-
p(W++1 2+1 /) p(W++1
H Causal CPC Pseudo algorithm
In this section, we present a detailed overview of the training procedure for Causal CPC. Initially, we train the Encoder using only the contrastive terms, as outlined in Algorithm [ Our primary objective is to ensure that, for each time step t, the process history H, is predictive of future local features Z.. However, calculating the InfoNCE loss for batch across all possible time steps 0, O,..., fmax can be computationally demanding.
To address this, adopt more efficient approach by uniformly sampling single time step t per batch. Subsequently, the corresponding process history H, is contrasted. The sampled H, is then employed as input for the InfoMax objective and randomly partitioned into future H and past H sub-processes.
The decoder trained while taking the encoder as input (Algorithm) 2. utilizing lower learning rate compared to the untrained part of the decoder. It is trained autoregressively and without teacher forcing. This implies that for each time step t, our GRU-based decoder should predict the future sequence of treatments Y Y++1t with its hidden state initialized the representation 4, of the historical process up to time
27

---

Algorithm Pretraining of the encoder
Require: Encoder parameters 2,,3,3, learning rate
Input: data {H i
......N] N}
for E (1..... maochmax do
for B H .mmaa L.....
B do
Z: = Lo, ((,,, 1,,, k-
fort= 0, mar.
Choose 2( maa
Compute C Lo, ,0 (H,)-
Compute _ CC (01, 02, (T, (Ty}T=))
Choose to ())
Compute C Lo, M, (H²), C. D (HH)
Compute ((InfoMan)) (01, 72
Update parameters
(C (1 + @c (0, (22.7) 00,,2,3
,1,2,3 t 01,2,3
00,,233
end for end for Return: Trained encoder.
To enhance training efficiency, instead of predicting Yic+1:t-- for all individuals i in a batch and for all possible time steps t, we randomly select time indices fi,1.... ft.m for each individual From these indices, we compute future treatment response sequences Y t7...., Yid mids, We found that is enough to train while selecting randomly 10% of the time steps.
Algorithm Training of the decoder
Require: Encoder parameters 0,,,3,, Decoder parameters 04, 0y, Ow.
Require: Encoder learning rate Penc: Treatment learning rate jaw Outcome learning rate jyy Number of random time indices m.
Require:
Input: data {H 1,6max 1,..., N]
for p E {l,.... maxch do
for B = {H, [Ha 1,
BB} do
Compute C,,: = encoder (H, for t = ..... tmans i I,.... [B].
Compute . o (H.).
for 1.... [B] do
Choose 4,,1 .. t. U([1 t mar -])- for / E (4,1,... 1i do Compute 1, W Wi.41147. T Pi4114tr11 decoder V., Waa Y,,, W Wit+lat) end for end for Compute ( 0y, Ow) and Lw (Ow, On).
Update parameters in the order.
iLdeo (OR,0y By, 0y)
0 1,2,33 01,2,3 Mene
001,2,3
(L (A Oy. By,yyy,mm (,,00w)
O1Y B1Y - py
00,,
[dLw (fw, OR)
Ow + Ow - pw
30w
end for end for Return:
Trained decoder.
28

---

I Causal CPC: Architecture details
Inputs: [x, 1111,
Linear Layer
WeightNorm
SELU
Linear Layer
WeightNorm
Outputs: Z, = 4Po, ([x,, Wi-1, Y--1))
Table 13: Architecture for learning local features Ze
Inputs: Ze
GRU (1 layer)
Outputs: Hidden state C, 00 (Zig)
Table 14: Architecture for learning context representation C,
Inputs: (e,, Wi]
Linear Layer
WeightNorm
SELU
Linear Layer
WeightNorm
Outputs: Y,
Table 15: Architecture for outcome prediction
Inputs: 4; Linear Layer
SpectralNorm
SELU
Linear Layer
SpectralNorm
Outputs: W, Table 16: Architecture for treatment prediction
Models hyperparameters
In this section, we report the range of all hyperparameters to be fine-tuned, as well as fixed hyperpa- rameters for all models and across the different datasets used in experiments. Best hyperparameter values are reported in the configuration files in the code repository.
29

---

Table 17: Hyper-paramen parameters search range for RMSN
Sab-medel
Hyperparamrter
Cancer simulation
MIXIC (NS)
LSTM avers
Letening rate
GOL, D.D., 0.00 D.IIO
0.01 BARO, CODIT JOKDI
Batch size
32,64, 128
32,
128
RMSM
Propensity Trestment Network
ES191 Trcalen unit
LSTXI dropout File
Max grallent nora:
1,2
1.
Early Suppping imin delks)
0.0001
0,0001
Farly Stopring (pseience)
30
30
Syers
D:00 1:000
0.01
66, (0000
Prupensity Histury Network
32,04, 128
ESTS
4.6..... 20
4.0.....
LSTM diopout rate
Early Stopping (min
0,0000
0.000
Farly Stupping (pwience)
LSTM lovers
Lesening rate
0O1, 000,0.005 0.005, 0201, 0.0000
0.01 0.005, (CO0T, 00000
Encoder
LSTM Inddes uniss
BalGh size
4.6. 207
02,04, 128
4.D..... 30
12, 128
LSTST Dispout rate
Farly Stepping (min dolla)
0.0001
0,0001
30
30
LSTM
0.01, 0.005,
0.01,
0.0001
Decuder
4.6.....27
4.0.....
LSTM diopout rate
gndieet norm
0,2
1.2
Early Stepping (min della)
0.0001
0.0001
Early Stopping (pasennc))
30
30
Table 18: Hyper- parameters search range for CRN
Model
Sub-model
Hyperpaameter
Cancer simulation
MUMIC (SS
LSTM liyers
0,01, 0.801 0.0001
0,00% 0.001
Ratch
128
CRN
Encoder
LSTM dropoul rate
1STM
4,0,
...00..0
4.6......
BR stre
4.0 20
4, 4,6. 30
Eany Stopping limin delia)
110001
010001
Early
(pstience)
30
30
Learning
Butch size
125, 256, 512
256. 512 1024
Decoder
LSTM Bi3son uniss
4,6, ....0..00
3, 3,
LSTM dropout rate
BR size
4,6
4.6.
Stopping dela)
0.0001
Early Suopping (palienue)
30
Table 19: Hyper- ppper-parameters search range for G-Net
Hyperparameter
Cancer simulation
MIMIC (SS)
LSTM layers
Learning rate
0.01, 0.005, 0.001, 0.0001
0.01, 0.005, 0.001, 0.0001
Batch size
32, 128
32, 64, 128
LSTM hidden units
4. G.6.....33 30
4,6. 30
FC hidden units
4, 9, 30
4, 30
LSTM dropout rate
size
G.....
6 ...... 30
MC samples
10
Early Stopping (min delta)
0.0001
0.0001
Early Stopping (paticnce)
30
30
Table 20: Hlyper-parametert search range for Causal Transfomer Hyperparameter
Cancer simulation
MI (SS)
Transfurmer blocks
Leariing
0.01, 0.005, 0,001, 0.0001 32, 64, 128 2 4. 1.6....21 20
0.01, 0.005 0.001, 0.0001 32, 64, 128 2
Batch size
Attention hcads
Transformer units
4, 1.6......00 20
I.TTM dropout rate
BR size
6, 20
6. 20
FC hidden units
4. 6.6..... 20
4,6, 6. 20
Scquential dropout rate
0.1, 0.2, 0.3
0.1, 0.2 0.3
Max positional encoding
15
Early Stopping (min delta)
0.0001
0.0001
Early Stopping (patience)
30
30
30

---

Table 21: seper-parameters search range for Causal CPC
Sub-andd
Hyperparaneler
MIMIC
GRL layess
eaming
0.01 0.066 0.001 0.001.0.000 0.0001
0.01 0.005 o.01,
0.0001
Rotch eine
128,
GRU hi3Sen unis
4,8 M.eeoeell
Cansal CPC
Eneader
GRU dropons rile
Local features (LP) size
4.6, 4.6
Consest Kepresenitiin (CR) size
4.B.oor221
4,6
27
Early Stopping (unin delin)
[000
0.001
Early Stopping (palicnes)
100
100
GKL layers
0.000
0.01
(.0000
Leaming rane (encuder luning)
Learning (ircaiment nerwork)
125
GRI hidlin unils
Decoder
GRU dropous Fale
BK siee
CW sute
GRU lapor& Ireat Eneodes)
GRU hikkln units (Trest Fnender)
Ft Taddes unils
4,0 20
Runlom time indices (m))
10%
10%
Early Stopping delia)
0.01
0.001
Stopping
Table 22: Hyper- perpme-prameeers search range for Causal CPC
Medell
Sub-modll
Heperparametrr
Cancerr simulation
MIMIC IIT (8S)
GRL layers
0.001
0.01 0.0. U.000
Buach
128
GRT hidlen
drggous
Teatures (LF)
4.0......3
Context Representtion (CR) size
Early Stogging uaa delia)
B001
0A001
Farly Stapping (paiieec)
100
100
GR layer
Leuning rile (dacoder W/o eeaiment soo-netwock)
U.D. dERA, EARD (ADO)
REDL, U.RA, GLOL EIIII
1 Leaming (encuder fine-tining)
0.001, 0,006, 0.000 0000005
0,01,
0.0005. 0.0001 0.00055 JU0L,
0OL 0.00%,
Dacch
32
92,04, 128
GRU hidien
Decoder
GRU deopont
Taysr Eneodey)
Encoder)
FC hidden uniis
4.6,
1......0
Random lime indiess (m)
10%
1056
Early Singping (min delin)
0001
0.001
Tarly Stopping (patiences
30
50
NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The paper introduces novel method combining RNNs with CPC for long-term counterfactual regression, leveraging MI objectives for efficient representation learning, and demonstrates state-of-the-art results on both synthetic and real-world data. These claims are substantiated by the detailed theoretical alnd empirical analyses provided in the paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]
31

---

Justification:
While Causal CPC excels at large horizon predictions, it does not outperform SOTA models on short-term predictions (Table T
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "I "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, c.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: Detailed proof's are provided in Appendix G
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: All the experimental details related to the training and evaluation protocol datasets descriptions [F. and F.I are provided. A detailed description of the Causal CPC
architecture provided in ! Pseudo-algorithms for both the encoder and the decoder are also provided in H Code is provided in the supplementary material.
32

---

Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel urchitecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily to reproduce that algorithm.
algorithm, the paper should make it clear how
(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g.. large language model), then there should either be way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. the case of closed-source models, may be that access to the model limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, described in supplemental material?
Answer: [Yes]
Justification:
Code is provided in the supplementary material at submission.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
hhipss.c//ublic/uuddds wtwp:////eooeSeeeeeoeeeooeo
www.obiiiiiiioooooioiiooo for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
33

---

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Models' hyperparameter search range is provided in Appendix P the selection method provided at the beginning of Section [ and remaining details about training and testing provided in the experimental protocol (Appendix D
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [No]
Justification: is computationally demanding compute errors bars for all neural network models in our benchmark. However, we reported the mean and standard deviation of metrics for each experiment, computed from multiple runs.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall
run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
34

---

Justification: We provided the computation resources used in the title of Table 2 as well the time of execution in the same table for cancer simulation data. similar table is provided in Appendix F.2 for MIMIC IIII data (Table T2)
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code Of Ethics
Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics Ethics/https:/
Question:
Answer: [Yes]
Justification; Authors acknowledge conducting research in conformity with the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. if there is special consid- cration due to laws or regulations their jurisdiction).
Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification:
An impact statement is included in Appendix
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deploymeent of technologics that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
35

---

11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification; Authors of models and datasets are appropriately cited in the paper in the intro- duction (T) and experiment (6) sections. Original owners of some model implementations are properly credited in our code.
Guidelines:
The answer NA means that the paper does not existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g.. website). the copyright and terms of service of that source should be provided.
If assets released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
Code for Causal CPC is provided in the supplementary material.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moodl as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
36

---

The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjeets
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/teview based the requirements of your country institution) were obtained?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
37


Reviews:
[{"review_id": "3S9N0Nl6j0", "reviewer": "Reviewer_mdqp", "summary": "The paper presents a novel approach to counterfactual regression over time, particularly focusing on long-term predictions. It introduces a method that leverages RNNs combined with CPC and InfoMax. This approach aims to capture long-term dependencies in data and improve computational efficiency without relying on complex transformer models.", "questions": "1. In the problem formulation, the treatment of W is assumed to be discrete, is it possible to extend it to be continuous and would such an extension violate the theoretical result?\n2. What is the performance of the approach when dealing with time series under different trajectory lengths and high-dimensional cases? \n3. How should the counterfactual treatment be chosen in the experiment? Randomly chosen from the support or do the authors have some rules?\n4. Could the authors explain why the proposed method 'excels at large horizon predictions, it does not outperform SOTA models on short-term predictions'?", "limitations": "The authors claimed in the Checklist.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Performance: The proposed model achieves superior performance in long-term prediction tasks compared to existing models.\n2. Theoretical foundations: The paper provides a thorough theoretical grounding for the proposed method, including a new information-theoretic perspective on representation balancing.", "weaknesses": "1. Complexity: While the method avoids transformers, the combination of CPC and InfoMax introduces its own complexity, which may pose challenges for implementation and replication. Also reported in Table 2 and Table 11, where the proposed method requires more training time with fewer parameters compared to the Causal Transformer and G-Net. \n2. Scope of evaluation: Although the evaluation is comprehensive, it could be expanded to include more diverse datasets to further validate the robustness and generalizability of the method. For example, 1) what is the total trajectory length in the synthetic experiment? I wonder about the performance and an additional experiment showing the performance under different trajectory lengths would be appreciated. 2) what is the dimensionality of the variables? I wonder about the performance when dealing with the high-dimensional case.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback! \n## Weaknesses\n\n### Complexity: \n\nIt is true that our model requires fewer parameters compared to SOTA models but has a relatively higher training time. However, several points mitigate this concern. The reported training time for CPC includes both encoder and decoder training. Once the encoder is pre-trained, the decoder training takes under 10 minutes, which is faster than the Causal Transformer and CRN. From a practical perspective, prediction time is also crucial. When estimating counterfactual responses, multiple trajectories per individual are generated, significantly increasing the dataset size. The Causal Transformer is designed to be trained using teacher forcing, which is disabled during testing, requiring multiple updates and batch reloads to perform counterfactual regression over multiple horizons. G-Net faces a similar issue, relying on time-consuming Monte Carlo sampling. However, our model is substantially faster than baselines at prediction time and outperforms them on large horizons, offering a good trade-off between computational efficiency and prediction quality.\n\n### Scope of evaluation and question 2:\n\nIn our cancer simulation data, the maximum sequence length is $60$, with a projection horizon of $10$, as noted in the paper. For the MIMIC III data, sequence lengths range from a minimum of $20$ to a maximum of $100$, varying due to common censorship phenomena in real applications. During training, a flag variable indicates active entries per individual, ensuring that the contrastive loss is computed only when the time step is \"active.\" The dimensionality of the variables was $4$ for the synthetic experiment and $74$ for the MIMIC semi-synthetic experiment.\n\nWe recognize the importance of additional experiments with varying sequence lengths. To further challenge our model, we reduced the sequence length to $40$ for synthetic data while keeping the forecasting horizon at $\\tau = 10$. We also decreased the maximum sequence length for semi-synthetic data from $100$ to $60$. As shown in Figure 1,2,3 for cancer simulation and Figure 5 for MIMIC III of the one-page PDF, the error evolution mirrors the core paper's results. Despite these challenges, our model continues to outperform most baselines in long-term forecasting.\n\nHowever, it is important to note that our model does not outperform others in short-term forecasting, a limitation observed even in additional experiments. Nevertheless, its strong performance in long-term predictions underscores its potential for applications where long-term accuracy is crucial. These additional experiments further validate the robustness and generalizability of our method across varying trajectory lengths.\n\n**Questions:**\n\n1. From a theoretical perspective, it is possible to extend the theory to handle continuous treatments by replacing the treatment classifier with a treatment regressor. Since we maximize the likelihood, there will be no change in the definition of equilibrium in Theorem 5.4. However, the results should be reported as $\\mathbf{H}\\_t \\perp W_{t+1}$ instead of conditioning on each treatment value. The significant changes would affect the proof of Lemma G.3, where we find the best treatment classifier given a fixed representation $q(W_{t+1} = 1 \\mid \\Phi(\\mathbf{H}\\_t)), \\dots, q(W_{t+1} = K \\mid \\Phi(\\mathbf{H}\\_t))$. For continuous treatments, we would need to find an infinite number of values $q(W_{t+1} = \\omega \\mid \\Phi(\\mathbf{H}\\_t))$. Rigorously, the standard optimization problem in Eq. (18) should be converted to a functional optimization problem, where the goal is to find the best continuous density function $q$. Under mild regularity conditions, variational calculus can be applied to the continuous version. Thus, Lemma G.3 can be extended, and the rest of the proof should follow easily by converting summations to integrals. We would be glad to add a formal extension to Theorem 5.4 in the appendix.\n\nFrom a practical perspective, continuous treatments would be represented as a single dimension in training, whereas discrete treatments are represented by a $K$-dimensional vector (one-hot encoding). This might pose a risk that the treatment information is not adequately captured during counterfactual predictions when it is continuous. One possible solution is to discretize the continuous treatments, which would make the problem more straightforward for our model.\n\n\n3. We detailed the generation rules in the Experimental Protocol (Appendix D). For the cancer simulation data, we generated counterfactual treatments similarly to CRN and Causal Transformer papers. Trajectories are generated with a single treatment per trajectory, while the treatment slides over the forecasting range to generate multiple trajectories. The idea is to choose the best time step to apply chemo or radiotherapy and not apply any treatment in subsequent steps to see how tumor volume will evolve. For MIMIC semi-synthetic data, trajectories are generated such that at each time step, treatment is randomly chosen from the support. Therefore, treatments can be applied at multiple time steps for one counterfactual sequence of treatments.\n\n4. Please see the global rebuttal.\n\nThank you for your insightful comments and suggestions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XWm8ILZ960", "reviewer": "Reviewer_pP4H", "summary": "This paper proposes a new method for counterfactual outcome prediction over time, with the goal of avoiding highly complicated models and expensive computation while achieving state-of-the-art prediction accuracy for time series with long-range dependencies. To this end, the presented method employs RRNs for long-range forecasting, rather than more complicated network architecture like Transformers, together with contrastive predictive coding (CPC) to capture long-range dependencies in the presence of time varying confounders and information maximization (InfoMax) to help with learning invertible representation for addressing the identification issue. Experiments with synthetic and semi-synthetic data are conducted to demonstrate the efficacy and superiority of the proposed method.", "questions": "1. Why a mapping of $\\textbf{C}_t$, instead of $\\textbf{C}_t$ itself is used as input of the decoder part of the proposed architecture?\n2. Could you provide any explanation on why the performance advantage of the proposed method starts to drop when $\\tao$ becomes bigger (as shown in the middle and right diagrams in Figure 3, and Table 10)? Moreover, why the baselines in general over outperform the proposed method with small horizon predictions?\n3. Table 2: the results shown are for the tumor growth data only and $\\gamma=1$. How the number of model parameters, training and testing time when different $\\gamma$ values are used?\n4. Why MSM is not used/reported in all the experiments? Having poor performance may not be a convincing reason for excluding it.", "limitations": "The paper has indicated some limitations of the proposed method, e.g. in Fig. 3 and Section 6.4, but more explicit discussions on the limitations in terms of decreased performance/superiority when the prediction horizon is small or large and when there are unobserved confounders should be given.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The aim to develop less complicated neural network models to tackle the challenging problem of counterfactual regression over time.\n2. The proposed approach is technically sound and the experiment result looks promising overall.\n3. The writing is very effective, and the authors have done a great job on clearly presenting the complicated process/components of their method.", "weaknesses": "1. The efficacy of the proposed method for long-range predictions is only demonstrated by experiments to certain extend, and from the results, it is not clear how the method would perform when $\\tao$ is greater than 10 steps. In particular, from the middle and right diagrams in Fig. 3, the advantage of the  proposed method over the baselines start to drop and some of the baselines start to outperform the proposed method. The paper would be much stronger if some theoretical evidence of the performance of the proposed method could be provided.\n2. There are some doubts with the experiment results or performance of the proposed method (please see the Questions section for more details).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your review!\n\n## Weaknesses\n\nWe conducted a similar experiment as suggested by the reviewer mdqp, where we reduced the sequence length seen during training but maintained a large forecasting horizon ($\\tau$). It is important to note that the middle and right diagrams in Figure 3 refer to a more challenging experiment with a high level of confounding. The main idea behind this experiment is that since the level of confounding is unknown in real data, it is beneficial to test different levels of confounding.\n\nFigures 1, 2, and 3 in the one-page PDF in our global rebuttal show that Causal CPC still outperforms in the new experiment with large forecasting horizons. It is true that in Figure 3, while Causal CPC outperforms in most forecasting horizons, its prediction for the last time step is close to that of CRN. However, this is only because $\\tau=10$ is the last forecasting horizon at which the process history was contrasted, and not specifically related to $\\tau=10$.\n\nTo further support our claims, we reran our model with the initial sequence lengths, and this time with $\\tau=15$ and $\\gamma=2$. Results in Figure 7 of the one-page PDF show that our model still outperforms baselines at forecasting horizons larger than $\\tau=10$ because the encoder is retrained in such a way that the InfoNCE loss is computed across all 15-time steps. We observe that the last prediction error is close to that of the baselines. This suggests that it may be desirable to train the model over larger forecasting horizons than initially intended.\n\n\n## Questions\n\n*Why is a mapping of $\\mathbf{C}_t$, instead of $\\mathbf{C}_t$ itself, used as input to the decoder part of the proposed architecture?*\n\n1. We use a mapping of $\\mathbf{C}_t$ to match the decoder's input dimension. This mapping acts as a dimension adapter, allowing the model to be more general. During our experiments, we kept the dimensions of both the mapping and the original representation consistent.\n\n*Could you explain why the performance advantage of the proposed method starts to decline as $\\tau$ increases (as shown in the middle and right diagrams in Figure 3 and Table 10)? Additionally, why do the baselines generally outperform the proposed method for short-term predictions?*\n\n2. The decline in performance with increasing $\\tau$ is due to high levels of confounding in the data, which affects all models. This confounding makes it challenging to maintain performance over longer horizons. For short-term predictions, the proposed method is designed for long-term predictions, which may explain why baseline models, optimized for immediate predictions, outperform it in the short term. This decline is observed primarily with the tumor growth dataset, **which has a low dimensionality of covariates (only 4 dimensions)**. However, our model performs consistently on more challenging datasets like MIMIC-III. We will add an experiment where we retrain the models to predict over more than 10 horizons.\n\n\n3. Due to space constraints, we have included similar results for the MIMIC-III dataset in Appendix F.2.3, Table 11. For the different versions of the cancer simulation according to $\\gamma$, there is no substantial increase in the training and prediction times. As we increase the level of confounding, the fine-tuning scheme tends to select a slightly higher dimension for the representation. For example, the dimension of the representation was 14 for $\\gamma=1$ and 16 for $\\gamma=2$. Overall, the increase in the number of parameters does not exceed $5\\%$ in the worst cases across all levels of confounding.\n\n4. MSM was excluded from Figure 3 to improve readability due to its high error values. However, we made sure to include it in the corresponding detailed table of results (Table 6 Appendix E.2.1). We will especially add MSM results to Table 1, which are the following: \n\n| Model       | $\\tau = 1$      | $\\tau = 2$      | $\\tau = 3$      | $\\tau = 4$     | $\\tau = 5$      | $\\tau = 6$      | $\\tau = 7$      | $\\tau = 8$      | $\\tau = 9$      | $\\tau = 10$     |\n|-------------|-----------------|-----------------|-----------------|----------------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|\n| MSM Table 1 | 1.20 $\\pm$ 0.10 | 1.83 $\\pm$ 0.26 | 2.07 $\\pm$ 0.40 | 2.3 $\\pm$ 0.45 | 2.54 $\\pm$ 0.45 | 2.90 $\\pm$ 0.37 | 3.01 $\\pm$ 0.38 | 3.06 $\\pm$ 0.37 | 3.08 $\\pm$ 0.36 | 3.09 $\\pm$ 0.36 |\n\n\n## Limitations\n\nAs suggested by reviewer PYKS, we will add a discussion of the limitations, particularly regarding the performance in short-term predictions, to the conclusion. Additionally, as you mentioned, we will emphasize this discussion in the falsifiability section. We intend to include at the end of line 358: \"We also observe in the falsifiability test, similar to our experiments, that Causal CPC does not particularly outperform baselines in short-term prediction, starting from $\\tau=6$. However, in Table 1, when sequential ignorability holds, our model starts to perform similarly to baselines at $\\tau=4$ and outperforms them at $\\tau=5$. This indicates that the violation of causal assumptions may relatively exacerbate the issue of not outperforming in short-term horizons.\"\n\nThank you once again for your feedback and comments.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7aIxi6hiqs", "reviewer": "Reviewer_PYKS", "summary": "This paper presents causal CPC, a framework for predicting counterfactual responses under time-varying treatments. The proposed method consists of two components: an encoder that leverages contrastive predictive coding and infomax principle to learn a representation of the history H, and a decoder that leverages a loss function based on adversarial game to predict outcome and treatment with balanced representations to account for selection bias. The concept of utilizing MI-based principles for casual representation learning in the longitudinal setting is interesting, whose efficiency has been demonstrated by the SOTA. However, I believe the paper could benefit from more streamlined writing with a clearer justification of how the use of MI-based principles enables efficient learning of long-term dependency.", "questions": "1. How to decide the relative weighting for the encoder and decoder during fine-tuning?\n2. Related to Weakness #2, how to better illustrate the role of L_infoNCE given the relatively mild drop in performance after removing it?", "limitations": "Limitations were mentioned in the appendix, but not much in the main text. Adding more discussion of model limitations in the Conclusion section might help.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The concept of utilizing MI-based principles for casual representation learning in the longitudinal setting is interesting.\n2. The performance of the model has been validated with a number of datasets, including fully-synthetic and semi-synthetic data.\n3. Some learning objectives are backed up with theoretical evidence.", "weaknesses": "1. Even thought the paper demonstrates the benefits of using CPC and infomax for learning long-term dependency with RNN, it still remains unclear to me, both in intuition and in methodological details, how this benefit was attained. The paper can benefit from some more insights, particularly in section 5.1, that links the proposed information theoretic methods to their actual benefits. The theoretical analysis seems a bit detached from the context and not so helpful in providing insights for the effectiveness of the proposed method.\n\n2. Following on the second point, in the ablation study, it seems that the removal of the InfoNCE loss is non-significant. This seems somewhat curious to me given the amount of emphasis the authors put on the importance of CPC.\n\n3. Some recent work in counterfactual outcome prediction with time-varying treatments should be discussed in the related work:\n - Berrevoets et al, Disentangled counterfactual recurrent networks for treatment effect inference over time \n\n - Chen et al, A Multi-Task Gaussian Process Model for Inferring Time-Varying Treatment Effects in Panel Data \n\n - Wu et al, Counterfactual Generative Models for Time-Varying Treatment \n\n - Frauen et al, Estimating average causal effects from patient trajectories", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your review!\n## Weaknesses\n\n**On the importance of the INfoNCE loss**: We have provided extended results pointing out the discrepancy in errors for all forecasting horizons in Table 8, specifically for the semi-synthetic MIMIC-III data. We reported only the average error over all horizons in the core paper due to space constraints.\n\nHowever, while computing the average error for the Full Model, we made an error by counting the error for the first horizon twice. In fact, from Table 8, **the actual average error of the Full Model is $0.62$ and not $0.66$**. To provide a clear understanding of the importance of regularization terms, we plot the error evolution of models similar to Figure 3 with the actual values of Table 8 **(Figure 4, the one-page pdf)**. The detailed ablation study in Table 8 shows that starting from $\\tau =2$, the drop in error when removing InfoNCE loss is consistently at $9\\%$.\n\nFor the synthetic dataset, we have verified that the reported average error based on the extended table of ablation (Table 7) is correct, and there are no typos. For the synthetic dataset, we should agree that the impact of the InfoNCE loss is still less significant. However, there are reasons related to the nature of the cancer simulation dataset itself, where the dimensionality of the time-varying component of the process history plus static covariates, $\\mathbf{U}\\_{t}=[\\mathbf{V}, \\mathbf{X}\\_{t}, W_{t-1}, Y_{t-1}]$, is relatively low, with **only four dimensions**. Our model is specifically designed to excel on datasets with a higher number of confounding dimensions, leveraging our modeling and regularizations based on contrastive losses. These findings align with our results on MIMIC-III, where the dimensionality of $\\mathbf{U}\\_{t}$ is **substantially larger at 72**, and our model consistently outperforms baselines at larger prediction horizons.\n\n**On the suggested recent work in counterfactual regression**: Thank you for your valuable suggestions. We will include the mentioned works in our related work section. However, while these papers provide interesting contributions to the field, we actually did not initially include them for the following reasons that we will gladly to related work discussion:\n\n- **Berrevoets et al. (2021)** focuses on sequences of binary treatments and requires a stronger version of the sequential ignorability assumption: sequential ignorability conditioning on current covariates. However, we assume a weaker version of sequential ignorability, which holds by conditioning on the entire history of the covariates, thus accommodating long-lasting confounding effects.\n\n- **Chen et al. (2023)** considers only binary treatments and targets the estimation of the average effect on the treated. More importantly, the authors assume a specific treatment assignment scenario where pre-treatment and post-treatment regimes can be defined for all treated individuals. In our paper's notation, this implies $W_{it} = 1$ for $g=1$ and $t > T_0$, with $T_0$ being the time of treatment application for all treated individuals and $g$ the index of the two groups (1 for treated, 0 otherwise). Our setting is more general, allowing for a complex assignment mechanism that varies individually and permits arbitrary swings in treatment value. Treatment and control groups vary substantially over time, and we have multiple treatment groups because the treatment is non-binary, making Chen et al. (2023) incompatible with our causal assumptions.\n\n- **Wu et al. (2023)** addresses high-dimensional outcome generation of counterfactuals according to a time-varying treatment plan, adhering to the same sequential ignorability assumption. However, it is not designed for causal forecasting over multiple time steps.\n\n- **Frauen et al. (2023)** is specifically tailored to estimating the average causal effect and cannot be used to estimate conditional counterfactual responses or individual treatment effects, as it targets the marginal expected counterfactual response using the g-computation.\n\n## Questions\n1. In fact, there is no relative weighting of the encoder and the decoder during fine-tuning. During pretraining, the encoder is trained using contrastive losses. At the fine-tuning stage, the decoder is trained from scratch using the adversarial game:\n\n$$\n\\begin{aligned}\n& \\min_{\\theta_{R}, \\theta_{Y}} \\mathcal{L}\\_{dec}(\\theta_{R}, \\theta_{Y}, \\theta_{W}) = \\mathcal{L}\\_Y(\\theta_{R}, \\theta_{Y}) + I_{\\text{CLUB}}(\\Phi_{\\theta_{R}}(\\mathbf{H}\\_t), W_{t+1}; q_{\\theta_{W}}) \\\\\n& \\min_{\\theta_{W}} \\mathcal{L}\\_{W}(\\theta_{W}, \\theta_{R}) = -\\mathbb{E}\\_{\\Phi_{\\theta_{R}}(\\mathbf{H}\\_t)} \\left[ \\log q_{\\theta_{W}}(W_{t+1} \\mid \\Phi_{\\theta_{R}}(\\mathbf{H}\\_t)) \\right]\n\\end{aligned}\n$$\n\nInstead of freezing the parameters of the encoder, we allow a slow update (small learning rate compared to the decoder) of its parameters following the losses defined in the adversarial game only. During the fine-tuning, there is no reconsideration of the encoder loss made of contrastive terms.\n\n## Limitations\n\nThank you very much for this comment. We will provide an updated version of the conclusion by including \"**While our model is tailored for long-term prediction, it does not outperform SOTA models for short horizons. However, our design of contrastive loss, especially InfoNCE in Eq. (4), suggests the possibility of deciding on a trade-off between short-term prediction quality and long-term prediction without training the model twice for the two objectives by designing suitable weights for each contrastive term at a given time step. We leave such an investigation for future work**.\"\n\nWe thank you again for your detailed review and valuable feedback, which helped correct a very important typo. If you have any more questions or need further clarification, please feel free to reach out.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "L81HrxAGMf", "reviewer": "Reviewer_6PYm", "summary": "The paper introduces a novel algorithm for long-term counterfactual forecasting over time through representation learning of historical information and balanced representation learning that predicts the outcome given balanced treatments. The representation learning process combines Contrastive Predictive Coding and Information Maximization, while the training of balanced representation learning involves an adversarial game in the actual outcome and treatment networks.", "questions": "1. In equation 1, $U_{t+j}$ in the numerator has one subscript, while $U_{l,t+j}$ in the denominator has two subscripts. Could you please make them consistent? Is $U_{t+j}$ information for one specific individual or an aggregate for all individuals? Is InfoNCE calculated for each individual or as the expectation over all individuals? Given that in equation 1, the subscript is $\\mathcal{B}$, is it the individual index\n2. Could you elaborate on how Theorem 5.2 influences the construction of the loss function in Equation 7? Specifically, does Theorem 5.2 impact the formulation or design of the loss function described in Equation 7? The authors discuss what we can infer if equality holds from lines 189 to 192, but will equality actually hold? The description in this paragraph suggests that equality can be achieved, yet Proposition 5.1 and Theorem 5.2 indicate positive gaps between the two mutual information measures without specifying the magnitude of these gaps, and the construction of the contrastive loss does not appear to be influenced by Theorem 5.2, as long as Proposition 5.1 holds.\n3. Could you explain what is $\\gamma$ in equation 7? Is $\\gamma$ a similar definition of horizons?\n4. In line 202, is there any proof provided regarding \"The theoretical existence of such a function\"\n5. Since the goal is to estimate the counterfactual responses given in the equation at the bottom of page 3, could you explain why we still need to predict the treatments as the conditional expectation is given $ W_{t+1:t+\\tau} $? Is this training needed because we want balanced learning? If so, could you explain why selection bias should be avoided given the goal of learning the expectation conditioning on potential treatments?\n6. In Theorem 5.3, could you explain why or when the “if” condition holds? \n7. Does the balanced learning only work for $W_{t+1}$? Why is it not necessary to extend it to $W_{t+\\tau}$ for $\\forall \\tau\\geq 1$?\n8. In section 5.3, is $\\theta_{R}=[\\theta_1, \\theta_2]$?", "limitations": "The authors have discussed the broader societal impacts.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper introduces a novel algorithm for predicting counterfactual responses over time which outperforms the baselines in both a synthetic and a semi-synthetic dataset.\n2. The proposed algorithm is robust. \n3. The paper is well-structured and comprehensive, covering theoretical analysis, experiments, ablation studies, and falsifiability tests.\n4. The experiment settings are described in detail.\n5. Most of the notation is clear and easy to understand.", "weaknesses": "1. Is it possible to conduct experiments on real datasets?\n2. It appears there is only one synthetic dataset and one semi-synthetic dataset. In the synthetic experiment, is the standard deviation of the errors calculated from experiments with 5 different seeds instead of random settings of the data generating process? Would it be feasible to conduct experiments on multiple synthetic datasets with random settings to obtain error bars, rather than relying on just one synthetic dataset?\n3. There don't seem to be any comparisons of time consumption with baselines.\n4. The loss functions are constructed based on the lower or upper bound of the mutual information, but there is no discussion or measurement regarding the tightness of the bound.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your feedback! \n\n## Weaknesses\n1. and 3. Please see the global rebuttal.\n\n2. Using multiple synthetic datasets with random settings to obtain error bars is a valuable suggestion, but we face constraints: limited budget for computational resources, increased energy consumption which conflicts with our goal to reduce carbon footprint, and access to only one cluster with a single GPU. Despite these limitations, our methodology remains robust and aligns with recent practices, such as those seen in the Causal Transformer.\n\n4. Measuring tightness for high-dimensional variables is challenging and computationally intensive, as discussed in [Rainforth et al., 2018; Poole et al., 2019]. This is typically feasible only with low-dimensional toy datasets or Gaussian assumptions. Batch size is crucial, as shown by the inequality in Eq. (3). Larger batch sizes $\\mathcal{B}$ tighten the lower bound by increasing $\\log(|\\mathcal{B}|)$. We used batch sizes of 256 for the encoder and 128 for the decoder to balance feasibility and performance. Despite not having perfectly tight bounds, mutual information and self-supervision provide beneficial inductive bias, as discussed in Appendix C.2. Our ablation study shows that replacing InfoNCE with estimators like NWJ, which has less bias but higher variance, did not improve performance. Our original model, using InfoNCE and InfoMax, outperformed other approaches (see Table 9, Appendix F2.1).\n\n## Questions\n\n1. $\\mathbf{U}\\_{t+j}$ refers to information for a specific individual. To simplify notation, we avoided using a subscript $i$ for individuals and only used double subscripts for batches, as in Eq. (1). We will add this note before discussing our modeling: \"To simplify notation, we remove the subscript $i$ from $\\mathbf{H}\\_{i,t}$ when discussing an individual, unless contextually necessary.\" Thus, the InfoNCE loss in Eq. (1) is the expectation over all individuals in the batch, denoted by $\\mathcal{B}$.\n\n2. The rationale behind the InfoMax Principle is to maximize the mutual information (MI) between the input process history $\\mathbf{H}\\_{t}$ and the learned context $\\mathbf{C}\\_t$. Before Proposition 5.1, we argued that maximizing MI between two representations—one for a historical subsequence $\\mathbf{H}\\_t^f$ (denoted $\\mathbf{C}\\_t^h$) and one for a future subsequence $\\mathbf{H}\\_t^f$ (denoted $\\mathbf{C}\\_t^f$)—is more beneficial both computationally and from an inductive bias perspective. We needed a theoretical justification for this approach, provided in Proposition 5.1, which shows it represents a lower bound of the original InfoMax term. Theorem 5.2 formally defines the gap between these terms; this gap is positive but can be zero theoretically. When the gap is zero (i.e., equality in Proposition 5.1), Theorem 5.2 indicates this occurs when the learned representations $\\mathbf{C}\\_t^h$, $\\mathbf{C}\\_t^f$, and $\\mathbf{C}\\_t$ satisfy $\\mathbf{H}\\_t \\perp \\mathbf{C}\\_t^f \\mid \\mathbf{C}\\_t^h$ and $\\mathbb{P}\\_{\\mathbf{C}\\_{t}^h \\mid \\mathbf{H}\\_t} = \\mathbb{P}\\_{\\mathbf{C}\\_{t}^h \\mid \\mathbf{C}\\_{t}^f}$. \n\nThe justification for the loss in Eq. (7) comes not from Theorem 5.2 alone but from its combination with Proposition 5.1, which shows our InfoMax simplification is a valid lower bound. Therefore, the contrastive loss in Eq. (7) is justified since we have $ \\log(|\\mathcal{B}|) - \\mathcal{L}^{(InfoMax)} \\leq I(\\mathbf{C}\\_t^h,\\mathbf{C}\\_t^f)\\leq I(\\mathbf{H}\\_{t}, (\\mathbf{C}\\_t^h, \\mathbf{C}\\_t^f)) $.\n\n3. In Eq. (7), $\\gamma$ refers to the parameters of the non-linear discriminator used in the definition of InfoMax loss.\n\n4. Yes, indeed! We have included proof in the appendix, specifically Proposition G.2. We will add a reference to it in the core paper.\n\n5. As you suggested, training is crucial to ensure covariate balance in the representation space. Our goal is counterfactual regression, specifically estimating $\\mathbb{E}(Y_{t+\\tau}(\\omega_{t+1:t+\\tau}) \\mid \\mathbf{H}\\_{t+1})$. For simplicity, let $\\tau=1$, our goal is to estimate:\n$$\n\\mathbb{E}(Y_{t+1}(\\omega_{t+1}) \\mid \\mathbf{H}\\_{t+1}) = \\mathbb{E}(Y_{t+1} \\mid \\mathbf{H}\\_{t+1}, W_{t+1} = \\omega_{t+1}) = f(\\mathbf{H}\\_{t+1}, W_{t+1})\n$$\nObserved data includes only one treatment regime per individual $i$ at each time step, $W_{i, t+1} = \\omega_{i, t+1}$. After fitting the regression model $\\hat{f}$, we estimate counterfactual responses for the same individual $i$ under different treatments, $\\hat{f}(\\mathbf{h}\\_{i, t+1}, \\omega_{t+1}')$ where $\\omega_{t+1}' \\neq \\omega_{i,t+1}$ (e.g., radiotherapy instead of chemotherapy). The challenge with treatment switching at inference is that $\\mathbf{H}\\_{t+1}$ and $W_{t+1}$ are not independent. This lack of independence can bias counterfactual estimates, introducing selection bias [Robins, 1999]. To address this, we learned a representation $\\mathbf{\\Phi}(\\mathbf{H}\\_{t+1})$ during decoding to remove selection bias.\n\n6. Theorem 5.3 highlights the importance of optimizing the treatment classifier: maximizing its log-likelihood leads to a smaller divergence $D_{KL}(p(\\mathbf{\\Phi}\\_{t+1}, \\omega_{t+1}) \\| q_{\\theta_W}(\\mathbf{\\Phi}\\_{t+1}, \\omega_{t+1}))$. Effective training of the treatment classifier helps meet the “if” condition, validating $I_{\\text{CLUB}}$ as an upper bound on MI between representation and treatment. This motivates our adversarial game framework, which alternates between optimizing the treatment classifier and minimizing the CLUB upper bound, as formally proven in Theorem 5.4.\n\n7. Covariate balancing in the representation space applies beyond $t+1$. For simplicity, we presented it for $t+1$, but in practice, it extends to all forecasting horizons. Our theorem holds for other horizons by replacing $\\mathbf{H}\\_t$ with $\\mathbf{H}\\_{t+\\tau -1}$ and $W_{t+1}$ with $W_{t+\\tau}$.\n\n8. Yes! We will add it right after Eq. (7).\n\nWe hope we have addressed your insightful questions and concerns. Thank you once again.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ht61nfs1JQ", "reviewer": "Reviewer_DSyv", "summary": "This paper leverages Contrastive Predictive Coding (CPC) with RNN for counterfactual regression over time to provide a compelling alternative to transformer-based approaches (which are challenging to interpret).  By leveraging CPC to capture long-term dependencies and InfoMax for \"reconstructable\" representation, the method achieves state-of-the-art results in counterfactual estimation.", "questions": "-  Line 178-179: *$H_{t}$ is a sequence of high-dimensional covariates and the computation of such loss is computationally demanding.* \n- Line 165-167: *...where at each horizon the discriminator parameter matrix $\\Gamma_{j}$ is of a small number of dimensions since being a map between two lower-dimensional representations*.\n\nLower bounds are motivated by the problem of high-dimensionality of $H_{t}$.\nWhat is the dimension of $H_{t}, Z_{t}$ and $C_{t}$ in the performed experiments?", "limitations": "See Weaknesses.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The method seems sound. \n\n- The paper is well-written. \n\n- Previous work leveraging contrastive learning for causal inference applies only to the static setting with no theoretical grounding. This paper frames the representation balancing problem from an information-theoretic perspective. The authors show that the suggested adversarial game yields theoretically balanced representations using MI's Contrastive Log-ratio Upper Bound (CLUB), computed efficiently.\n\n-  Achieved good empirical performance with fewer patient data.  Good ablation studies.", "weaknesses": "- Experiments: performance in random trajectories for the pharmacokinetic-pharmacodynamic model of tumor growth is missing. \n\n- While good results are achieved with fewer patient data, it is important to check if the results are consistent as the patient number increases. \n\n- Line 96: \"*...the role of invertible representation in improving counterfactual regression. Here, we introduce an InfoMax regularization term to make our encoder easier to invert.*\" The authors provide some intuitions (lines 204-213) in favor of invertibility (beyond reconstructable) but they are not sufficient. A formal proof is needed. \n\n- For a “reconstructable” representation of the process history $H_{t}$,  While a lower bound of InfoMax objective ( regularization) is optimized during pre-training of the encoder, there is no guarantee that finetuning of pre-training will retain the desired “reconstructable” property of the representation.\nHence, the role of invertible (or reconstructable) representation in improving counterfactual regression is not well understood even though ablation studies show improvement when $\\mathcal{L}^{Infomax}$ is introduced.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your review !\n\n## Weaknesses\n\n1. We have included the performance evolution on random trajectories for the pharmacokinetic-pharmacodynamic model of tumor growth in Figure 3 of the core paper. This figure illustrates the model's performance across multiple levels of confounding to evaluate its robustness. The precise values of the errors for each confounding level and at each time step are detailed in Appendix E.2.1, Table 6, due to space constraints in the main paper.\n\n2. In Appendix F.2.2, we have actually increased the number of patients and tested our model in the same setting of train/validation/test as [Melnychuk et al. (2022)]. The Causal CPC still gives better results than SOTA at the majority of the forecasting horizons.\n\n\n3. Thank you for highlighting this important point. We acknowledge that a formal proof would strengthen the argument for the role of invertibility in improving counterfactual regression. Here, we provide a more formal basis for the claim.\n\nTo ensure identifiability in the latent space, we leverage recent advances in causal and disentangled representation learning. Suppose the true data-generating process is given by $\\mathbf{H}\\_{t} = g(\\mathbf{z}\\_t)$, where $\\mathbf{z}\\_t$ represents the true latent factors. In the sequential context, we assume that the same function $g$ generates two historical subsequences:\n$$\n\\mathbf{H}\\_t^f = g(\\mathbf{z}\\_t^f), \\quad \\mathbf{H}\\_t^h = g(\\mathbf{z}\\_t^h)\n$$\nWe assume a general dependency of the form:\n\n$$\np(\\mathbf{z}\\_t^f \\mid \\mathbf{z}\\_t^h ) = \\frac{Q(\\mathbf{z}\\_t^f)}{Z(\\mathbf{z}\\_t^h)} \\exp(-d(\\mathbf{z}\\_t^f, \\mathbf{z}\\_t^h))\n$$\n\nHere, $\\mathbf{\\Phi}$ is an encoder, and we use the InfoMax regularization term as follows:\n\n$$\n\\mathcal{L}^{(InfoMax)}(\\mathbf{\\Phi}, d, \\mathcal{B}) := -\\mathbb{E}\\_{\\mathcal{B}} \\left[ \\log \\frac{\\exp(-d(\\mathbf{\\Phi}(\\mathbf{H}\\_t^f), \\mathbf{\\Phi}(\\mathbf{H}\\_t^h)))}{\\sum_{l = 1}^{|\\mathcal{B}|} \\exp(-d(\\mathbf{\\Phi}(\\mathbf{H}\\_{l,t}^f), \\mathbf{\\Phi}(\\mathbf{H}\\_t^h)))} \\right]\n$$\n\nAccording to Matthes et al. (2023), under certain conditions, if the encoder $f$ minimizes $\\mathcal{L}^{(InfoMax)}$, then $h = g \\circ f$ is a scaled permutation matrix. This result suggests that when the encoder achieves a minimizer for $\\mathcal{L}^{(InfoMax)}$, the encoder function $f$ closely approximates an invertible transformation of $g$.\n\nFrom a causal inference perspective, if $Y_{it}(\\omega_{it}) \\perp W_{it} \\mid \\mathbf{H}\\_{it}$ and $\\mathbf{H}\\_{it} = g(\\mathbf{Z}\\_{it})$, then an invertible function $g \\circ f$ ensures that:\n\n$$\nY_{it}(\\omega_{it}) \\perp W_{it} \\mid g \\circ f (\\mathbf{H}\\_{it})\n$$\n\nThus, $Y_{it}(\\omega_{it}) \\perp W_{it} \\mid g(\\mathbf{C}\\_{it})$ and since $g$ is invertible, we have:\n\n$$\nY_{it}(\\omega_{it}) \\perp W_{it} \\mid \\mathbf{C}\\_{it}\n$$\n\nThis demonstrates that the representation $\\mathbf{C}\\_{it}$ retains the essential independence structure, facilitating accurate counterfactual inference.\n\n\n4. During the decoder training, the encoder is fine-tuned. However, to ensure that the encoder is only \"slowly\" optimized during the fine-tuning, we actually choose a learning rate of $5.10^{-4}$, whereas, for the decoder, we ensure a faster convergence by using a ten times higher learning rate of $5.10^{-3}$.\n\n## Questions\n\n1. For the cancer simulation data: $\\dim(\\mathbf{H}\\_t) = 4$, $\\dim(\\mathbf{Z}\\_t) = 12$, $\\dim(\\mathbf{C}\\_t) = 14$. For the MIMIC III data: $\\dim(\\mathbf{H}\\_t) = 74$, $\\dim(\\mathbf{Z}\\_t) = 14$, $\\dim(\\mathbf{C}\\_t) = 14$.\n\nYour comment highlights a crucial aspect of our research field. In fact, obtaining suitable datasets to verify counterfactual methods, particularly those applied over time with a considerable number of covariates, is a significant challenge in the causality research field. While benchmark datasets for causal inference in static settings are relatively more abundant, as detailed in the survey by Yao et al. (2021), longitudinal datasets that fit our framework are rare.\n\nGiven the increasing interest in causal inference, it is our hope that more longitudinal datasets, especially those with relatively high-dimensional covariates, become publicly available. Such datasets would significantly benefit the research community by providing more opportunities for validation and comparison of different methods.\n\nDespite these challenges, we conducted experiments aligned with baseline methods [Lim et al., 2018; Bica et al., 2020; Melnychuk et al., 2022] using both the Tumor Growth dataset and the challenging MIMIC III dataset. Our results showcased superior performance in long-term counterfactual regression, demonstrating the effectiveness of our proposed method.\n\nOnce again, we thank you for your insightful comments and hope our responses address your concerns. We welcome further discussion or questions to clarify any remaining issues.\n\n[Bica et al., 2020] Bica, I., Alaa, A. M., Jordon, J., and van der Schaar, M. (2020). Estimating\ncounterfactual treatment outcomes over time through adversarially balanced representations.\n\n[Lim, 2018] Lim, B. (2018). Forecasting treatment responses over time using recurrent marginal struc-\ntural networks.\n\n[Matthes et al., 2023] Matthes, S., Han, Z., and Shen, H. (2023). Towards a unified framework of\ncontrastive learning for disentangled representations.\n\n[Melnychuk et al., 2022] Melnychuk, V., Frauen, D., and Feuerriegel, S. (2022). Causal transformer\nfor estimating counterfactual outcomes.\n\n[Yao et al., 2021] Yao, L., Chu, Z., Li, S., Li, Y., Gao, J., and Zhang, A. (2021). A survey on causal\ninference.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ex2SlNqKhu", "reviewer": "Reviewer_Nj8s", "summary": "the paper proposes a method based on CPC to estimate the causal effect of treatments over a period of time", "questions": "not many, mostly how susceptible is the method to violations of the assumptions the authors have set out ?", "limitations": "adequately discussed", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Interesting paper\n- well written , although some things could have been cleaner \n- good experimentation and ablation study \n- interesting use of cpc", "weaknesses": "- Not clear from an intuition perspective if the contrastive approach incentivises the model to learn what the authors want \n- The complexity of the method is significant , raising questions about applicability and reproducibility \n- The point about invertible processes is not a great one as invertible processes severely limit the function space available without offering much in return. They also increase computational complexity significantly\n - not the first use of a contrastive objective in causal learning , id suggest that this contribution is toned down a bit", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your review!\n\n## Weaknesses\n\n1. Please see the global rebuttal.\n\n2. We have reported the complexity of our model and that of baselines for experiments on both synthetic and semi-synthetic data. For instance, in Table 2 of the core paper, we show the training time of our model (aggregated encoder and decoder) and the prediction time for counterfactual trajectories. We provide similar details for MIMIC III data in Appendix F.2.3, Table 11. Although our model has fewer parameters compared to SOTA models, its training time for both the encoder and decoder is comparable to SOTA, even though some baselines, such as the Causal Transformer and G-Net, consist of a single end-to-end trained model.\n\nWe employed several techniques to achieve efficiency without sacrificing performance, detailed in Algorithms 1 and 2 for encoder and decoder training.\n\n**Trick 1:** Since computing the InfoNCE loss for all time steps $t = 0, \\dots, t_{\\text{max}}$ can be computationally intensive, we sample a single time step $t$ per batch, using the corresponding process history $\\mathbf{H}\\_t$ for the InfoMax objective. The sampled $\\mathbf{H}\\_t$ is then partitioned into future $\\mathbf{H}\\_t^f$ and past $\\mathbf{H}\\_t^h$ sub-processes.\n\n**Trick 2:** The decoder is trained autoregressively and without teacher forcing. For each time step $t$, our GRU-based decoder predicts the future sequence of treatments $\\hat{Y}\\_{t+1:t+\\tau}$ with its hidden state initialized to $\\mathbf{\\Phi}\\_t$. To enhance training efficiency, we randomly select $m$ time indices $t_{i,1}, \\dots, t_{i,m}$ for each individual $i$ and compute future treatment response sequences $\\hat{Y}\\_{i,t_{i,1}+1:t_{i,1}+\\tau}, \\dots, \\hat{Y}\\_{i,t_{i,m}+1:t_{i,m}+\\tau}$. It is sufficient to train using only 10% of the time steps.\n\n\n3. The point about invertibility discussed just before Section 5.2 considers the implicit inversion of the encoder. Recent literature on contrastive learning suggests that an encoder-only architecture, when trained with contrastive loss, implicitly approximates an inverse of the true data-generating process. Our argument is that our encoder, trained with the InfoMax loss, might benefit from this byproduct without additional cost. Thus, there is no need for a decoder to reconstruct the input space, potentially reducing computational complexity. To address concerns about invertibility, we have added formal proof to this assertion in response to Reviewer DSyv.\n\n\n4. In lines 103-114 of our paper, we discuss the novelty of applying the InfoMax principle to causal inference, specifically for time-varying data. Our work extends contrastive learning beyond static settings, as seen in [Chu et al., 2022] and [Zhu et al., 2024]. We provide theoretical arguments showing how selection bias is completely resolved in the representation space, a critical aspect not fully addressed in these static settings. We appreciate your feedback and will consider it carefully to further position our work accurately within the existing research landscape.\n\n## Questions\n\n1. The crucial assumption in our work is sequential ignorability, which is necessary for identifying conditional counterfactual responses. This assumption implies no unobserved confounders, whether static or time-varying. We tested our model's robustness by training Causal CPC and baselines on MIMIC III data with some confounders masked. Results are reported in Section 6.4, Table 4 of the core paper. Compared to Table 1, where assumptions are not violated, errors for Causal CPC, Causal Transformer, and CRN increase when confounders are masked, except for RMSN, which remains insensitive during the robustness test. However, RMSN starts to underperform significantly at $\\tau \\geq 2$. Our model continues to outperform baselines in long-term forecasts even when sequential ignorability is violated, demonstrating its ability to encode long-term dependencies of observed confounders effectively.\n\nThank you once again for your valuable feedback.\n\n[Chu et al., 2022] Chu, Z., Rathbun, S. L., and Li, S. (2022). Learning infomax and domain-\nindependent representations for causal effect inference with real-world data.\n\n[Zhu et al., 2024] Zhu, M., Wu, A., Li, H., Xiong, R., Li, B., Yang, X., Qin, X., Zhen, P., Guo, J.,\nWu, F., et al. (2024). Contrastive balancing representation learning for heterogeneous dose-response\ncurves estimation.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
## Global Rebuttal

We would like to first thank the reviewers for their very constructive feedback and valuable comments.

1. We can use real data, like the MIMIC III dataset, instead of its semi-synthetic version. However, evaluating counterfactual trajectories is not possible due to the absence of counterfactual responses in real datasets. Upon the request of reviewer 6PYm, we assessed our model and baselines by forecasting factual responses over time and estimating responses for each individual's observed treatment trajectory. Our experiments on real MIMIC III data, shown in Figure 6 of the 1-page PDF, demonstrate that our model consistently outperforms all baselines at large horizons, highlighting the effectiveness of our model design.

2. To further consolidate the discussed intuition in lines 204-213 regarding the implicit inversion of the encoder, going beyond the reconstructability of the representation classically advocated by the InfoMax principle, we provided a proof sketch upon the request of reviewer DSyv in favor of this claim.

3. Following the suggestion of reviewer mdqp, we added additional experiments where we reduced the sequence length from 60 to 40 for synthetic data while keeping the forecasting horizon at $\tau = 10$. We also decreased the maximum sequence length for semi-synthetic data from 100 to 60. As shown in Figures 1,2,3, and 5 of the one-page PDF, the error evolution mirrors the core paper's results. Our model continues to outperform most baselines in long-term forecasting.

5. We would like to further emphasize the intuitions behind the design of the InfoNCE loss, as raised by reviewers Nj8s and PYKS, which can also explain why our model excels at large prediction horizons but not in the short term, a question raised by reviewers pP4H and mdqb. Our main intuition is to learn a representation of $\mathbf{H}\_{t+1}$ that is highly predictive of the future components of the process, i.e., $\mathbf{U}\_{t+j}=[\mathbf{V}, \mathbf{X}\_{t+j}, W_{t+j-1}, Y_{t+j-1}]$ for $j=1, \dots, \tau$. To achieve this, we learn a representation of the process history leveraging an RNN, referred to as the context $\mathbf{C}\_t$, and aim to make it predictive, at large horizons, of different future local representations $\mathbf{Z}\_t$ of process components $\mathbf{U}\_{t+j}$. We ensure such predictiveness using the InfoNCE loss $\mathcal{L}^{(InfoNCE)}\_j$, for all the prediction time steps $j=1, \dots, \tau$.

**To encourage the context $\mathbf{C}\_t$ to learn the shared information between all future local representations, particularly future covariates**, we minimize the InfoNCE loss averaged across all future time predictions via $\mathcal{L}^{CPC}:= \frac{1}{\tau} \sum_{j = 1}^{\tau} \mathcal{L}^{(InfoNCE)}\_j$. We can also trivially write as given in Eq. (5):

$$
\frac{1}{\tau} \sum_{j = 1}^{\tau} I(\mathbf{U}\_{t+j}, \mathbf{C}\_{t}) \geq \log(|\mathcal{B}|) - \mathcal{L}^{CPC}.
$$

Therefore, as we minimize $\mathcal{L}^{CPC}$, **we push the model to learn a context representation that shares the maximum information with the future components across all prediction time steps.** This approach encourages the model to capture the **global structure of the process (when $\tau$ is large)**, which is very beneficial for performing counterfactual regression over large horizons. It is, therefore, expected, given the tailored design for long-term prediction, that the model does not necessarily outperform SOTA models at short-term horizons. We further demonstrated the importance of the InfoNCE loss using an ablation study.

6. Finally, we compared the time consumption of our model with baselines in Table 2 of the core paper for a synthetic dataset with a confounding level ($\gamma = 1$), including both training and prediction times. The same table for the semi-synthetic dataset is in Table 11 of Appendix F. 

Thank you for your insightful comments and suggestions.

Meta-review: The paper presents a novel approach to counterfactual regression over time, i.e., long-term predictions. It combines RNNs with CPC and InfoMax. This approach aims to capture long-term dependencies in data without relying on complex transformer models. There is a general   consensus among the reviewers that the paper contains interesting ideas and will make non-negligible contributions to the field. The reviewers raised several important concerns, most of which were addressed in the authors' rebuttal. It should be possible for the authors to address the remaining concerns in the camera-ready version of the manuscript.

Hence, I recommend that the paper be accepted as a poster.
Decision: Accept (poster)