CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals

Tomàs Garriga; Gerard Sanz; Eduard Serrahima de Cambra; Axel Brando

cs.LGFeb 17, 2026

CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals

Authors: Tomàs Garriga, Gerard Sanz, Eduard Serrahima de Cambra, Axel Brando

Organizations: Novartis · Barcelona Supercomputing Center

Abstract

The ability to accurately perform counterfactual inference on time series is crucial for decision-making in fields like finance, healthcare, and marketing, as it allows us to understand the impact of events or treatments on outcomes over time. In this paper, we introduce a new counterfactual inference approach tailored to time series data impacted by market events, which is motivated by an industrial application. Utilizing the abduction-action-prediction procedure and the Structural Causal Model framework, we first adapt methods based on variational autoencoders and adversarial autoencoders, both previously used in counterfactual literature although not in time series settings. Then, we present the Conditional Entropy-Penalized Autoencoder (CEPAE), a novel autoencoder-based approach for counterfactual inference, which employs an entropy penalization loss over the latent space to encourage disentangled data representations. We validate our approach both theoretically and experimentally on synthetic, semi-synthetic, and real-world datasets, showing that CEPAE generally outperforms the other approaches in the evaluated metrics.

Figures & tables

Figure 1: Example of a time series counterfactual (cf) estimation comparison for our synthetic dataset among a baseline LSTM-based forecast model and CEPAE. The vertical line separates pre-event from post-event observations, counterfactual estimations and ground truth.

Panel of Figure 2 — Figure 2: Problem setting’s computational graphs. Solid arrows denote direct influences on $\bm{Y}$ ; dotted double arrow reflects the abduction link. The confounded panel adds $\bm{H}{\rightarrow}E$ .

Figure 3: Scheme of encoder-decoder based methods for counterfactual inference, both in training and counterfactual inference phase.

Model	Regularizer on $\bm{Z}$	Practical implications
CVAE	KL divergence with fixed prior $p(\bm{z})$	Stable training with standard VAE machinery, but often poor reconstruction and insufficient disentanglement between $\bm{Z}$ and $\bm{C}$ , which can hurt counterfactual accuracy.
CAAE	Adversarial training on $\bm{Z}$ using an event discriminator.	Encourages disentanglement between $\bm{Z}$ and $\bm{C}$ and has good reconstruction, but inherits the instability and tuning difficulties of adversarial training.
CEPAE	Entropy penalty $S(\bm{Z})$ .	Simple objective (no discriminator); encourages disentanglement between $\bm{Z}$ and $\bm{C}$ and has good reconstruction, having a stable training.

Table 1: Summary of the three encoder–decoder models used for counterfactual estimation. Here

\bm{C}

denotes the conditioners (e.g.,

(\bm{h},e)

Ground-truth metrics (synthetic and semi-synthetic only)
		Synthetic		Conf. synth.		Semi-synth.		Real world
Metric	Method	0	1	0	1	0	1	0	1
cf MAE $\downarrow$	LSTM	$.199\pm.005$	$.198\pm.005$	$.199\pm.003$	$.201\pm.004$	$.101\pm.004$	$.080\pm.002$	–	–
	AB-LSTM	$.201\pm.006$	$.197\pm.005$	$.202\pm.009$	$.202\pm.011$	$.103\pm.003$	$.083\pm.003$	–	–
	CVAE	$.144\pm.021$	$.137\pm.014$	$.145\pm.007$	$.187\pm.017$	$.105\pm.005$	$.083\pm.004$	–	–
	CAAE	$.087\pm.012$	$.083\pm.011$	–	–	$.065\pm.004$	$.061\pm.003$	–	–

Table 2: Combined results across datasets for

e_{f}{=}0

and

e_{f}{=}1

(10 seeds). We evaluate both intervention directions: setting 0 where

e_{f}{=}0

and

e_{cf}{=}1

, and setting 1 where

e_{f}{=}1

and

e_{cf}{=}0

. Top block: ground-truth metrics (cf MAE/cf MBE; only available on synthetic and semi-synthetic datasets). Middle: Added Variations (Total/Altered Steps). Bottom: axiomatic metrics (Reconstruction, Reversibility, Effectiveness). “–” indicates not applicable (no counterfactual ground truth) or not reported due to unstable training.

Figure 4: Synthetic Control and Causal Impact comparison.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

$\sigma$	MAE
$0.0$	$0.004$ 0.002
$0.1$	$0.045$ 0.002
$0.3$	$0.077$ 0.001
$1.0$	$0.114$ 0.002
$1.5$	$0.128$ 0.002

Appendix

Table 3: MAE of the synthetic-control baseline for different noise standard deviations

\sigma

Figure 5: Convergence of CEPAE with a fixed EP weight of $0.09$ on the unconfounded semi-synthetic dataset. We plot counterfactual MAE versus training epoch. The model converges slowly but reaches a stable low-error region around epoch 100.

Figure 6: Convergence of CEPAE with a linear schedule for the EP weight on the unconfounded semi-synthetic dataset. The EP weight increases linearly from $0$ to $0.09$ during the initial epochs. Counterfactual MAE decreases more rapidly than in the fixed-weight case and stabilizes around epoch 50.

Figure 7: Convergence of CAAE with a linear schedule for the adversarial weight on the unconfounded semi-synthetic dataset. The adversarial weight increases from $0$ to $8.9$ during training. After an initial drop in counterfactual MAE around epoch 50, the loss slowly increases again, illustrating the instability inherited from adversarial training.

Figure 8: Example of a failure mode for CAAE under a linear adversarial-weight schedule. At some point during training, the counterfactual MAE suddenly increases and remains high for the rest of the epochs, showing a typical instance of adversarial instability.

Figure 9: Convergence of CAAE with a fixed adversarial weight of $8.9$ on the unconfounded semi-synthetic dataset. Without a schedule, the model fails to reach low-error regions and counterfactual MAE remains high throughout training.

Metric	Latent dimensionality $D_{z}$
	6	7	8	9	10	11	12
MAE	$0.062\pm 0.004$	$0.058\pm 0.003$	$0.056\pm 0.004$	$0.051\pm 0.003$	$0.049\pm 0.004$	$0.055\pm 0.004$	$0.051\pm 0.004$
MBE	$-0.008\pm 0.002$	$0.002\pm 0.003$	$0.002\pm 0.004$	$0.006\pm 0.004$	$0.010\pm 0.003$	$0.006\pm 0.002$	$-0.003\pm 0.002$

Appendix

Table 4: Sensitivity of CEPAE to the latent dimensionality

D_{z}

on the unconfounded semi-synthetic dataset. We report counterfactual MAE and MBE (mean

\pm

one standard deviation across seeds). The EP weight is fixed to

0.09

Metric	EP weight $\lambda_{\mathrm{EP}}$
	0.05	0.07	0.09	0.11	0.13	0.15
MAE	$0.055\pm 0.004$	$0.051\pm 0.003$	$0.056\pm 0.004$	$0.056\pm 0.003$	$0.060\pm 0.004$	$0.075\pm 0.004$
MBE	$0.011\pm 0.004$	$0.004\pm 0.003$	$0.003\pm 0.004$	$0.004\pm 0.004$	$0.003\pm 0.002$	$0.003\pm 0.002$

Appendix

Table 5: Sensitivity of CEPAE to the EP weight

\lambda_{\mathrm{EP}}

on the unconfounded semi-synthetic dataset. We report counterfactual MAE and MBE (mean

\pm

one standard deviation across seeds). The latent dimensionality is fixed to

D_{z}=8

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Counterfactual estimation over time is important in various applications, such as personalized medicine. However, time-dependent confounding bias in observational data still poses a significant challenge in achieving accurate and efficient estimation. We introduce causal autoencoding and treatment conditioning (CAETC), a novel method for this problem. Built on adversarial representation learning, our method leverages an autoencoding architecture to learn a partially invertible and treatment-invariant representation, where the outcome prediction task is cast as applying a treatment-specific conditioning on the representation. Our design is independent of the underlying sequence model and can be applied to existing architectures such as long short-term memories (LSTMs) or temporal convolution networks (TCNs). We conduct extensive experiments on synthetic, semi-synthetic, and real-world data to demonstrate that CAETC yields significant improvement in counterfactual estimation over existing methods.

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