CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals
Authors: Tomas Garriga
Gerard Sanz
Eduard Serrahima de Cambra
Axel Brando
Venue: Accepted by TMLR ()

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 arises from an industrial context. Utilizing the abduction-action-prediction procedure and the Structural Causal Model framework, we begin employing methods based on variational autoencoders and adversarial autoencoders, both previously used in counterfactual works 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 achieve disentangled data representations. We validate our approach both theoretically and experimentally on synthetic, semi-synthetic, and real-world datasets, showing that CEPAE outperforms the other approaches in the evaluated metrics.

Published in Transactions on Machine Learning Research (February/202g
CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals
Tomàs Garriga
tomas.gorrigo. tomas.gorrioo_ddcuzzzoott dicuzzo@novartis.ccm
Novartis
Borcelona Supercompuling Center
Gerard Sanz
Novartis
Eduard Serrahima de Cambra Novartis
edward.sermhhim wwmbro.@novrrtss.com
Axel Brando
azell brendo &bsc.es
Borcelona Supercompuling Center
Reviewed on OpenReview: https: // openreview. net/ forum? id- X6lrzg0t@o
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 under- stand the impact of events or treatments on outcomes over time. 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 Ooduction-action prediction procedure and the Structural Causal Model framework, first adapt methods based variational autoencoders and adversarial autoencoders, both previously used in counterfactual literature although not in time series settings. Then, we present the Condi- tional Entropy-Penalized Autoencoder (CEPAE), a novel autoemcoder-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 theoreti- cally and experimentally on synthetic, semi-syuthetic, and real-world datasets, showing that CEPAE generally outperforms the other approachos in the evaluated metrics.
1 Introduction
Time series counterfactual estimation is essential tool understand an event's impact time series data. Tt is applied to situations where an event or treatment at certain point in time alters time series' trajectory, and consists in inferring, once given the observed post-event data, the counterfactual, i.e., the time series that would have taken place if the event had not occurred. Common applications of time series counterfactuals include finance (Barocas et al.) 2020) or healthcare (Prosperi al. 2020). Although our focus here is on thcoretical and empirical aspects, the method was originally developed for an application in the pharmaceutical industry, where counterfactuals are used for business planning. Pharmaceutical companies are notably affected when drug's patent expires and Loss of Exclusivity (LOE) (Castanheira et al. 2019
takes place, prompting competitors to launch cheaper generic versions of the drug. This usnally results in a dramatic decrease in the sales volume of about 60-70% in the first years (Castanheira et al. 2019 severely ingecting the company's revenues. Thus, accurately assessing the market impact of generic drug entries is of great importance. See appendix A for an extended discussion on the industrial application of this work.
Synthetic control methods are widely used for time-serics counterfactual estimation (Bouttell al., 2018
Abadie & Gardeazabal 2003). For example, Causal Impact (Brodersen et al. 2015) estimates counterfactuals

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..* Ground truth of Forecasi eelinate Opserved time series
Figure 1: Example of a time series counterfactual (cf) estimation comparison for our synthetic dataset among baseline LSTM-based forecast model and CEPAE. The vertical line separates pre-event from post-event observations, counterfactual estimations and ground truth.
from two information sources: pre-event observations of the target series and control series that predict the target and are assumed unaffected by the intervention. These methods require suitable control series at inference time. In our industrial setting, such controls are often unavailable or only weakly predictive. A proxy for
common alternative is to train forccasting model on historical data and use its predictions the counterfactual, but this ignores information contained in the observed post-event trajectory.
In this paper, we introduce novel approach for time-series counterfactual estimation that leverages post- event observations of the larget series while not requiring any additional time series at inference time, i.., it extracts post-event information directly from the observed target trajectory. It is rooted in recent advances in causal machine lcarning (Pawlowski et al.) 2020), leveraging the abduecion-actiio-prrdddi procedure (Pearl 2000) and encoder-decoder architectures. We instantiate this framework with three en- coder-decoder models. First,, adapt Conditional Variational Autoencoder (CVAF), which has been widely used for counterfactual estimation in image and tabular domains. Second, consider Conditional methods. Finally, we introduce Conditional Entropy-Penalized Autoencoder (CEPAE), which uses an entropy penalty on the latent space to encourage disentanglement from the conditioners. To the best of our knowledge, CEPAE is a new architecture for coun- ual estimation. Figure shows an observed time series that suffers, after the initial drop associated to an event, an additional drop which not related it; we see that CEPAE recovers the additional drop in the counterfactual estimate, whereas the LSTM forecast baseline does not. Although our approach requires sufficient amount of event and event-less time series to train the models, which can be limitation, it has the advantage of not needing control time series for inference. Having successfully applied our method in our industrial context to infer impacts of generic drug competitors in pharmaceutical industry, in this work we present various experiments that demonstrate its validity.
Adversarial Autoencoder (CAAE) inspired by
This paper is struct as follows: Sec. offers a review of the existing literal ure on counterfactual inference. Sec. provides the necessary background on Structural Causal Models (SCMs) and information theory. Sec. details our methodology, covering the utilized causality concepts, the problem setting and the explanation of our models. In Sec. 5 we introduce the datasets, models and metrics, and discuss the results. Finally, in Sec. G we discuss the conclusions.
Our main contributions are: 1) An adaptation of the abduecioon-cction-rreee predictonnn procedure, which has typically been used with other forms of data such as images, lo a time series setting. We have adapted, as well, two autoencoder-baseel models: CVAE and CAAE, 2) The introduction of CEPAE, an entropy penalized autoencoder for counterfactual estimation, demonstrating its validity both theoretically and experimentally.
2 Related Work
Our models are inspired, on the one hand, by SCMs and the abduection-cction--rddiie procedure [Pearl 2000), and, on the other hand, on autoencoder (AE) (Bank et al., (2023) and variational autocncoder (Kingma & Welling 2013) architectures. Parafita & Vitria (0099 proposes method for leveraging deep learning
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prowess to estimate counterfactuals using the aforementioned Judea Pearl theory. Pawlowski et al. (2020) develops similar method that specifically uses VAFs. Other related approaches that use autoencoder based architectures are Maachez-Martinn et al. (2021), which use variational graph autoencoders, or Kim et al.
(2021), which proposes a VAE-based approach that clusters the causal graph. In appendix we provide an extended related work that covers other counterfactual approaches that are not based on autoencoders.
On the other hand, most counterfactual estimation works for time series are based on the aforementioned synthetic control and related approaches, which are also covered in appendix B
It is important to notice that there several works in the Explainability field that use the term counterfac- tual in completely different sense than this work. In the Explainability context, if we consider, for example, a binary classifier and a given input, the term counterfactual refers to the most similar input to the one given that delivers a different classifier outcome. This approach can help understand how the classifier works, but is different from the causal concept of counterfactuall Some works that tackle the Explainability counterfac- tual problem are applied to time series (Wang et al., 2023 2021]. However, it is important to recognize the difference between this approach and the causal counterfact ual problem that our work addresses.
Finally, within the causal inference literature, there is line of research, represented by works like (Bica et al, 2020 Melnychuk et al. 2022), that addresses a problem called time varying counterfactuals, but the word counterfactual there does not follow the J. Pearl definition, and this line tackles the problem of forecasting outcomes under different temporal sequences of treatments, not the problem of inferring the time series that would have taken place if an event had not occurred.
3
Background
In this section, we provide an overview of SCMs and information theory, which are essential for our approach.
3.1
Structural Causal Models
The proposed approach for counterfactual estimation is based on the J. Pearl definition of counterfactual Pearl 2000), which can be operationalized by employing SCMs.
A SCM M = P(U)) consists of a collection S = ( of structural assignments 0; Se(ui; pa;). where pa, is the set of parents of a; (its direct causes), and joint distribution P(U) I P(U,) over mutually independent exogenous noise variables (i.e. unaccounted sources of variation) (Pawlowski el al. 2020). In structural causal model (SCM), in general, the assignments are assumed acyelic. Thus, a directed acyclic graph (DAG) can represent relationships, with edges pointing from causes to effects in causal graph. A unique joint observational distribution (M( is determined by every SCM, fulfilling the causal Markov assumption: each variable is independent of its non-effects given its direct causes. Thus, it factorizes as (a) = II (PM(a; pa,)), where each conditional distribution (a, pa,) is determined by its assignments and noise distribution (Peters et al. 2017).
SCMs allow to perform counterfactual queries a three-step procedure (Pearl 2000 1) Abduction:
predict the 'state of the world' (the exogenous noise u) that is compatible with the observed data a, i.e. infer PM a); 2) Action: perform an intervention (e.g. do(A; ā,)) to the counterfactual SCM
which corresponds to the desired manipulation, which generates the modified counterfactual SCM A. := M a, (S, PM a)); 3) Prediction: compute the quantity of interest basod on the distribution entailed by the modified counterfactual SCM, (a)
3.2 Information Theory
For a random variable X, we denole as S(X) its entropy, or differential entropy when X is continuous. Considering an additional random variable Y, we denote the conditional entropy of X on Y as S(X and the mutual information among X and Y as I(X;Y). In appendix C these quantities are properly defined for both discrete and continuous variables, and we discuss some issues regarding differential entropy interpretation. From now on, we will not distinguish among entropy and differential entropy.
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For three random variables X, Y and Z, present four information theoretic properties, demonstrated in appendix C that are essential for our development:
S(X)
(1)
S(X,Y) +(X) + S(Y ((x;)),
(2)
S(X,Y) - +
(3)
- +
(4)
4
Method
In this section, we explain our problem setting, the encoder-decoder approach for counterfactual estimation, and the models that we use.
4.1 Time Series Counterfactuals: Problem Setting
Next, we define the structure of the time series setting over which our models perform counterfactuals Problem Statement. Let X = [=) be a set of observed time series x² (zf. 237) with T steps, with T = T1 + Tg- Some of these time series are impacted by an event e, at time step Tj 11 Our sctting implicitly assumes regularly sampled time series, i.c., constant time interval between consccutive observations. Time series that are not impacted by the event are also assigned a value that represents the absence of impact. Thus, we can divide each time series into a pre-event segment h, = (h.,,..., har, ) with Ti steps and a post-event segment = (Va...., .Vr,)) with T2 steps. From now on, we will denote as H, E and Y the variables referring, respectively, to the pre-event time series, the event and the post-event time series, and as h, e and y to their specific realizations, often distinguishing among factual values (ey and ys) and counterfactual values (Eef and yes). For an observation { hh' eyy, 38 our objective is to estimate the counterfactual values yog in the hypothetical scenario where had taken a different value (E = e while the rest of the factors of variation of Y were maintained.
Given the previously defined variables, define counterfactual function f such that yef = f(h, e ys, Then, our task is find an approximate counterfactual function f that,, similarly to f, estimates Def-
In our company, we have access to large amount of historical sales volume data for many products and countries, a significant amount of which have been impacted by event. Thus, from these data, can build time scries dataset of selected number of steps T which consists of non impacted time series, obtained by applying a T' steps rolling window to the non impacted historical data, and impacted time series, obtained by taking, once selected a number pre-event steps T₁ and post-event steps T2 (with Ti + T2 = T), the windows that match this selection in the impacted historical time series.
In our counterfactual setting, E and H will be the parents of Y, and we consider datasets with two structures: H being parent of F, becoming confounder (confounded setting), and direct relation existing among H and E' (unconfounded setting). In our counterfactual setting, E and H are always intervened on (H to its factual value h and E to the counterfactual value ect). Therefore, it is not necessary to treat E and H as random variables generated by exogenous noise terms that must be abducted. This makes counterfactual estimation in both the confounded and the unconfounded settings equivalent at a practical level. Figure 2 shows the computational and the causal graph of our problem. For the models that we present in this work, we assume that our variables data generating process follow a causal graph like the one shown in figure with no hidden confounders, and also positivity.
4.2 Encoder-Decode Based Models for Counterfactual Estimation our Problem Setting
Roadmap. We first describe a generic encoder docoder instantiation of pnnnetiionnactiiopprriciieeeeeee (Eq. 5 and Box 1). then summarize how CVAE and CAAE implement it, and finally introduce CEPAE and the entropy penalty that enforces the required independence between Z and the conditioners.
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H
H
Y
Ug
Y
U
E
E
(a) Unconfounded
(b) Confounded
Figure 2: Problem sctting's computational graphs. Solid arrows denote direct influences on Y; dotted double arrow reflects the abduction link. The confounded panel adds H -.
Post-event (4r)
Post-event reconstruction (
Of estimation (Ve))
Pre-event (h)
Encoder
Decoder
Cf inference
Training
ecf
Both
Figure 3: Scheme of encoder-decoder based methods for counterfactual inference, both in training and counterfactual inference phase.
For counterfactual estimation in nigh-dimensional settings, autoencoder- based architectures are natural choice as they provide flcxible latent representations (Pawlowski et al.) 2020 Sanchez-Martin et al., 2021 Lample el al. 2017). In this work, we adapl two such archilectures to our limeesseies setting and propose a new one. Next, we describe how the counterfactuals of our problem setting can be estimated using encoder- decoder architectures and assuming the graphs in figure
Considering the to define two trainable functions that, together, allow us to estimate a counterfactual def once given h, yf. ef and Eaf- We will refer to the first of these functions as an encoder Es, with trainable parameters o, which outputs either the location and scale parameters of a (usually multivariate) latent distribution PZZ h, 9f) Fig (h, ef: yf); or directly latent variable z F. Z is a. latent representation that should allow the sccond of our trainable functions, which we call the decoder Dg with trainable param- eters 0, to estimate counterfactuals D.f Dz(z, h, e.y). Figure 3 illustrates this process. In terms of the express functions jef f(h es, My, eef) Do(EE(h, ey. h.r)))
procedure and our counterfactual problem setling, it is possible
approximated counterfactual function mentioned in 4.1
(5)
In the case where the encoder outputs the parameters of the latent distribution, the value z will be ob- tained via sampling from that distribution. For quick reference, we summarize the counterfactual inference procedure in Box
Given (h, @h, el .vf) and coun- uerfactual event value esf; (i) Abduction: <- Eg eh, , (or sample ~ po(= h, el, w (ii) Action: replace ef by eaf keeping fixed; (iii) Prediction: ge + De(
Box 1. Method in three steps (ablucttor-actioo-prciitoo
There a direct analogy between this encoder decoder setting and the abduction action prediction pro- cedure. In SCM terms, abduction amounts to inferring a distribution over the exogenous noise U, given (h, yr), i.c., PM h.e,, wr). Our encoder plays the role of learning a latent variable that sum- marizes this noise: it aims to represent the part of the variation in that not explained by (h,
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Table 1: Summary of the three encoder decoder models used for counterfactual estimation. Here C denotes the conditioners (c.g., (h, e)).
Model
Regularizer on Z
Practical implications
CVAE
KL divergence with fixed
Stable training with standard VAE machinery, but often poor reconstruc- tion and insufficient disentanglement bet Z and C, which can hurt coun- terfactual accuracy.
prior p(z)
CAAE
Adversarial training on Z using an event discrimina- tor.
Encourages disentanglement between Z and C' and has good reconstruction, but inherits the instability and tuning diffi- culties of adversarial training.
CEРАΕ
Entropy penalty S(Z).
Simple objective (no discriminator); courages disentanglement between Z
and C and has good reconstruction, having a stable training.
Importantly, following impossibility results the identifiability of generative mechanisms with multidimen- sional exogenous variables Nasr-Esfahane & Kiciman 2023). the true exogenous noise Ug. is not identifiable in general. Therefore, we do not claim that Z equals Uy; instead, we only require that Z is a useful proxy that behaves like such a noise term for the purpose of counterfactual prediction. Then, the decoder (that can be identified with the structural assigument- fy) inferring Gef would correspond simultaneously to the action and the prediction steps, where we first set the modified SCM M = M and then the result Def is obtained as sample of M.
Why disentanglement matters. If we train a conditional autoencoder only with reconstruction loss, the latent variable Z can encode all the variation in y, including the effect of (H, E). In that case, the decoder may ignore the conditioners and, at test time, changing E will have little effect on the output. From the SCM in Fig. 2 and the independence of exogenous noises in Sec. 3.1 the variation in y can be decomposed into contributions from its parents HH, E) and exogenous noise term U, satisfying U, H E, want the decoder to use (H,E) to model
and U, 1 E. For counterfactual queries that intervene the causal effect of the event and to use Z only for the remaining variability. Intuitively, this corresponds to learning a representation Z that is (approximately) independent of (H, E) and captures only the variation analogous to Uy. In practice, this motivates regularizing the latent space so that cannot redundantly encode information already present in the conditioners.
The entropy penalty in CEPAE is designed precisely encourage this behaviour: Section 1.2.3 provides informationn-therreiic arguments showing that minimizing the entropy of Z tends reduce its mutual information with the conditioners while preserving information to reconstruct the input.
Next, we explain three autoencoder based models for counterfactual estimation: based model very common, for example, in image counterfactual inference; CAAE, a model inspired by existing works for image manipulation; and CEPAE, a novel counterfactual model which we show that overcomes the previous ones our time series sottings. We only briefly discuss the two first models here, and leave additional information for the appendix. Also, we summarize the core differences between these models in table 1
CVAE, an autocncoder
4.2.1 CVAE
We first apply a conditional VAE, with KL Hall 1987 regularization term, to infer counterfactuals ac- cording the previously defined process. See appendix for more details. Being CVAE reasonable option to generate counterfactuals, some relevant problems exist. For example, the model ignore, completely

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partially, its conditioning C, as even if KL divergence limits the capacity of Z to encode information, the model lacks theoretical guarantees of disentanglement of Z with respect to C. Some of the techniques that can be used to reduce the capacity of Z, such as increasing the weight of KL or reducing the dimensionality of latent space, have severe consequences on the reconstruction capabílities, which also affect the quality of the counterfactual estimations. Even without these techniques, the reconstruction capacity of VAEs is often poor, espocially if we compare to regular AEs (Zhao et al. 2017
4.2.2 CAAE
As mentioned above, regular autoencoders lack mechanisms to disentangle the latent representation from conditioners. On the other hand, their reconstruction capacity far exceeds VAEs. Thus, CAAE is first attempt to maintain the reconstruction capacity of AEs while furthering disentanglement properties. We take the idea from Lample et al. (2077), work focused on image manipulation, and adapt it to our counterfactual setting. Based on ideas from domain adapiation (Ganin et al.] 2016), the method consists in a conditioned AE where an adversarial training is added that makes the latent representation unpredictive of the value of the conditioner C, trying to achieve Z PL C. The adversarial training consists in a model that predicts the value of C from Z while the encoder parameters & are trained to achieve the opposite objective. For that, we use gradient reversal layer (Ganin et al., 2016), with an increasing linear scheduling as the weight for the adversarial objective. Alomar et al. [2023 uses a similar idea for counterfactual inference. See appendix E for more details on this model.
Adding adversarial training for several different conditioners can be difficult, especially if they are multidi- mensional like H. In our case, we apply the adversarial training only on E, which, in theory, should be enough for our objectives as, for our counterfactuals, we do not intervene on H.
In our experiments, CAAE performs well in the unconfounded setting but fails to produce accurate counterfactuals under confounding, despite extensive hyperparameter search. Existing adversarial image- manipulation methods similarly focus unconfounded settings and do not directly address our causal setup. Moreover, CAAE inherits the usual drawbacks of adversarial training, including instability and higher com- putational cost (Sridhar et al. 2021;; we illustrate these convergence issues empirically and compare them with the behaviour of CEPAE in Appendix M
4.2.3 CEPAE
We propose a conditional autocncoder based model that adds, to the regular reconstruction loss of an AE, an entropy penalty (EP) over the latent representation that reduces its entropy S(Z), minimizing the amount of information that Z can carry.
Intuitively, if the conditioners arc informative about X (i.c., I(X;C) > 0), then encoding this information in Z increases the entropy S(Z) without improving the joint information I(X; Z, C) that reconon depends on The entropy penalty therefore discourages Z from redundantly encoding information already available in C. Next, we provide a thcoretical foundation for this intuition.
Following eq. 1. we can decompose the joint entropy of Z and C in terms of their mutual information with X in the following way:
C) = 10: C* + S(Z, CIX).
(6)
On the other hand, from eq. 2 we can also dccompose S(Z,C) in the following way:
S(Z.C) S(Z) S(C) C)Z
(7)
Mixing both and isolating S(Z), we obtain:
S(Z) = [(X: C) -(C)
(8)
The term S(Z. C in the last equation
be decomposed, following eq. S(C, z = ((C + (z X). Finally, we obtain:
C|X)
S(Z) = I(x: Z, C) + S(Z I(: S(C S(C).
(9)
The reconstruction loss aims increase (X:X). and X (the reconstruction X) is lunction of Z and C.
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In the last equation, the terms S(C and S(C) are fixed from the dataset and do not depend on the model. The term T(X; Z, C)) would tend to be minimized by the entropy penalty alone, but the reconstruction loss counteracts this by increasing it. Properly weighting the entropy penalty ensures that the influence of the reconstruction loss prevails. The term S S(ZIC, X) can be interpreted as a noise term (the amount of entropy of Z which not predictive of X). Finally, T(Z;O) is the key term that we aim to minimize by penalizing S(Z). Like ((Z(,,) X), this term will be minimized with opposition exerted by the reconstruction loss and, since I(Z; C) has an inferior limit in 0, the penalization will tend to cancel it, thus aiming for Z 1 C. In a deterministic model, [(X; Z, C) and S C(Z can tend, respectively, towards infinity and minus infinity. In appendix [F] we further discuss on that and show how this does not affect the meaning and implications of the equation.
Implementation of the entropy penalty. The existing methods for estimating the entropy of continuous variables from a finite number of samples involve complex calculus and usually training specific models. To avoid chis, we derive simple expression that sets an upper bound for the entropy of a continuous multivariate distribution based on the variance of its variables. From equation 2 and taking into account that mutual information can not be negative, we have that S(Z) = S(Z,, Zx) S S(Z1) + + S(Zx), where N is the dimensionality of Z. On the other hand, there is a theorem that states that, for a single variable, the normal distribution maximizes the differential entropy for a given variance (see the demonstration in appendix C.1 or in Marsh )2013)). Thus, the entropy of a variable X ~ WN o²), which is S(X) = log(av 2Te). sets an upper bound to the entropy of an arbitrary distribution with variance 0². This gives us that S(Z) S n logo(v22mm
log(a)) + ... + log(ox) =: (Z)( where Stn(Z) is an upper bound to the entropy of Z based on 01, ... NN. Although Sub(Z) (Z) could, in theory, be used is unstable due to the logarithm tending to minus infinite for values near zero. We have tried adding small fixed values to the standard deviations inside logarithms but, although some improvements are achieved, model training continues being unstable, slow and suboptimal. Leveraging the fact that any way of reducing standard deviations will reduce SUB(Z), we have found that a simple summation over them is more stable and effective as a loss function. Thus, for a batch of data {a*, c sk=1 with B samples, our EP loss is:
entropy penalty loss function, it
N
Chatch
Σ 0
(10)
EP
1=1
where og is the standard deviation of the ith dimension of the batch of latent represontations {Z*) where z = { 1 and z = (x* e²).
On the other hand, the reconstruction loss is:
Ce (0, (0 = |105 Do (FFF)()),c)),
(11)
Then, for a batch of size NA, the loss of CEPAE is:
Na
CEPARE (0. 1
Croe (0, の) + X.hbttch (d).
(12)
NB
As with CVAE and CAAE, the framework described in beginning of 4.2 allows to generate a counterfactual sample Des by encoding an observation وز and its parents h and ey. Here, we have z = Eg h ef), and then jof= = Dø h ecf).
Like CAAE, CEPAE has the reconstruction power of regular AEs, which is a clear advantage over CVAE. On the other hand, it has some important advantages over CAAE. First of all, we avoid the instability of adversarial training, using instead an EP to achieve disentangled representations, which is a much simpler regularization technique regarding both computational demands and implementation difficulty. Additionally, although not leveraged in this work, EP implementation offers the advantage over adversarial training of not increasing in complexity with the number of conditioners or their dimensionality.
EP is somewhat reminiscent of the Information Bottleneck theory (Tihbyy et al., 2000), espocially to Strouse & Schwab (2017). although we do not closely follow this framework and the derived disentanglement prop- erties are alien to it. We further discuss this relationship in appendix G On the other hand, it is important
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to mention that works like Nasr-Esfahamn & Kiciman (023)) provide an impossibility result for identifia- bility of generation mechanisms with multidimensional exogenous variables, which affects all counterfactual works applied to multidimenniooall data. Nonetheless, as Nasr- Esfahany & Kiciman (2023) notes, "exact counteractual identifiability is often too strong, c.g., in cases where low counterfactual error is tolerable by praclitioners". Otherwise, all works on usunterfactualss in multidimensional data, like Pawlowski al. (2020)
or Sanchez & Tsaftaris (2022). should be ruled out by the same argument. As we show in the Experimental
Section (5), CEPAE's counterfactual errors are low, and deciding if they are tolerable will depend on each application. See appendix H for extended discussion.
5 Experimental Section
In this section, we explain the datasets and the metrics that we have used to validate our models, the details of the models, and finally the results.
5.1 Datasets
To evaluate the effectiveness and robustness of our methods, we use several time-series datasets with dif- ferent characteristics. In causal counterfactual evaluation, real-world datasets rarely provide ground-trun truth counterfactuals, direct error metrics often unavailable. Two common alternatives are: (i) using metries that assess desirable properties of counterfactuals rather than similarity to ground truth (Sec. 5.2 and (ii) using synthetic or semi-synthetic datasets where the data-generating process is (partly) controlled, enabling ground-trutie counterfactuals and standard error metrics. Accordingly, besides our proprietary dataset, we use synthotic and semi-syuthetic datasets that mirror our setting of abrupt impact-driven changes while providing counterfactua ground truth. All datasets are split into series with an event at a fixed time step (e (e=1) 1) and series without an event (e 0). We describe them below.
Synthetic datasets. We generate length-30 series starting at 0 with lincar trend sampled uniformly from [-0.1, 0.1] (added at each step). Series with an event include a drop of 0.7 at step 20. In addition, all series (with or without event) undergo an extra change at a random post-event step, sampled uniformly from [-0.7, capturing other shocks unrelated to the main event. We add Gaussian noise with variance 0.1 at each step. The only systematic difference between event and non-event series is the drop at step 20, which allows generating paired factual and counterfactual outcomes. We consider two variants: (1) Unconfounded, where event assignment is random; and (2) Confounded, where event probability follows Bernouli(p) with p= (t 0.1)/0.2, where is the trend.
Semi-synthetic dataset. Based on the Rossmann Store Sales dataset (Knauer & Cukierski 2015) (daily sales, 2013-2015), we simulate an event occurring on the first Monday of March cach year that affects half of the stores (e.g,, aromolion/markelinng markkeigg)... The event multiplies sales by 1.1 on day 1, 1.2 on day 2, and 1.3 for the remaining days over three weeks. We use the four weeks before the event as pre-event context and the following three weeks as post-cvent. This dataset is unconfounded.
Real-world dataset. Our private dataset contains monthly sales time series constructed described in Sec. 4.1 We use 12 months pre-event and 30 months post-event. The dataset is univariate and includes 2310 series with an event and 3000 without. According to business experts, the relationship between H and E' is non-existent or very weak.
5.2 Evaluated Metrics
To evaluate methods, use various metrics. Mean Absolute Error (MAE) and Mean Bias Error (MBE) are employed to compare estimations with ground truths in synthetic and semi-synthetic datasets, as counterfactual ground truths exist solely for these datasets.
MAE a direct measure of the accuracy of our estimates and is the most important metric, as it directly compares the counterfactual estimate with the ground truth. On the other hand, MBE, defined as MBE n - yi) allows to detect biases. We also consider additional metrics that do not require a ground truth, and hence can be applied to the real world dataset. Fach of these metrics assesses some desirable
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property of counterfactuals. Overall, these metrics allow us to evaluate counterfactual models even when ground-truth counterfactuals are not available. For completeness, we apply them to all datasets and not only to the real world one. Next, we present the Added Variations metrics, and leave the reconstruction, reversibility and effectiveness metrics from the Axiomatic Definition of Counterfactual (Monteiro et al. 2023 for appendix I due to space restrictions.
Added Variations. We introduce this metric to evaluate how a counterfactual estimation responds to changes in the observations.
The core idea is that, for an accurate method, if we introduce variations in the post-event factual time series, these changes should be treated as effects of processes unrelated to the event and therefore should be reflected in the counterfactual estimate. This metric is implemented as follows: for each time series to be evaluated, several positive and negative values in the order of the data values are chosen; for each of this values, several windows of few consecutive steps from the post-event time series are selected and the chosen value is added to those steps. After that, a counterfactual estimate is obtained for every altered time series and it is compared to the counterfactual estimate of the non-altered time series. Two quantities are obtained: 1) Total difference, that takes into account the difference among the altered counterfactual and the base counterfactual in all the steps, and 2) Altered steps difference, which takes into account the difference among the altered counterfactual and the base counterfactual only in the steps affectod by the alteration. These quantities are then divided by the expected difference (the product of the alleration value and the number of affected steps). Thus, ideally, the final results for both total difference and altered steps difference metrics should be 1. The final results are obtained by averaging all calculations. For a more formal definition of these metrics, see Appendix J
Axiomatic Metrics. Following Monteiro et al. 2023 and the Pearlian axioms of counterfactuals [Pearl 2000 Galles & Pearl 1998 Halpern] 2000). we summarize three model-agnostic metrics that assess coun- terfactual soundness without ground-truth counterfactuals. Let x be an observation with factual parents pa, and x² its counterfactual under parents pa". Denote the idcal counterfactual mapping by f, such that x² = [r. pa, pa"), and its learned approximation by f.
Reconstruction (Composition; lower is better). Null interventions should not change x. We measure (z, [z, f(z pa, pa)), with d₂ the Mean Absolute Error (MAE).
Recon := =d
Reversibility (lower is better). Counterfactual mappings should invert when swapping parents: Rev == dz (x. M((., pa. pa"), pa", pa))
Effectiveness (higher is better). Intervening parent pag to pa should be reflected in the gen- erated counterfactual. Using a pseudo-oracle Pas that predicts pas from an observation, we report Eff := d (Pay(fu pak. paj)), pai): with dş as accuracy for discrete parents (or & for continuous ones).
In our setting, x is the post-cvent time scries; factual parents are the pre-event series and factual event (h, ey), while counterfactual parents replace ef by eaf (keeping h). See appendix | for a more complete description.
5.3 Models and Baselines
We compare CVAE, CAAE and CEPAE for all the metrics described in 5.2 and we add as a benchmark, for the MAE and MBE comparison with ground truth counterfactuals, an L-TM-based conditional forecast model that has as inputs only H and E. Thus, it can be used as time series counterfactual estimator that does not take into account post-event values. Besides, we add Adversarially Balanced LSTM (AB-LSTM), which can be seen as an adaptation of the models like Counterfactual Recurrent Network (CRN) Bica et al. 2020) or Causal Transformer (CT) (Melyychuk et al.) 2022) to our setting. See appendix K for more details. Metrics like Added Variations only make sense for mothods that use the onddectio-prrdiitioonattii procedure. We omit CAAE metrics from the confounded synthetic dataset because, as mentioned in 4.2.2 it has not been possible to obtain reasonable results.
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Although synthetic control methods would be alternative to our approach, their reliance the quality of control data does not allow a fair comparison with the SCM based models, which do not use control data. In 5.4 we perform an illustrative synthetic control experiment with the synthetic dataset, where we see that it will or will not outperform CEPAE depending on how predictive the control data are. We see that, in each application, the selection of the optimal model should strongly depend on the control data quality, case exists.
The encoder and decoder architectures of CVAE, CAAE and CEPAE are shared, and are based on 1D convolutional and transposed convolutional layers, in setting inspired by the VAE model for time series generation proposed in Desai et al. (2021). All mothods have been implemented with TensorFlow (Abadi et al. 2015). using an Adam optimizer (Kingma, 2014) with a learning rate of 10-4. Other hyperparameters of CVAE, CAAE and CEPAE such as dimensionality of latent space or the weight of their respective regular- izations are particular for each dataset and have been chosen after an optimization process on factual data. For more details about implementation, the code is provided in https: / /tt//gittuu com/tgarriga/CCPP CatrrigaaaCEPEEEE. Also, appendix M provides additional convergence plots for CEPAE and CAAE, and appendix reports hyperparameter sensitivity analysis for CEPАE.
5.4 Results
Table 2 summarizes results across all datasets and both intervention directions (setting 0: ey=0 - eef=1; setting 1: ey=1 Ee)=0), averaged over 10 seeds. MAE/MBE are only available when counterfactual ground truth exists; hence "_" on the real-world dataset. Figure E reports the complementary synthetic- control comparison under varying donor predictiveness for the synthetic dataset.
Key takeaways.
Accuracy (ground-truth datasets): CEPAE achieves the lowest of MAE across all datasets and both intervention directions.
CEPAE roduces bias relative to CVAE and is best or comparable among abduction-action- prediction models (CVAE/CAAE/CEPAD)
Bias:
Proxy metrics (all datascts): CEPAE consistently closest the idcal Addod Variations behaviour and achieves the best or near-best axiomatic scores.
Practicality: removing entropy penalization causes severe degradation (ablation), and CEPAE Irains faster than CAAE/CVAE under our setup.
Synthetic control: when strong donors exist, synthatic-controlssyyle basclines can match or out- perform CEPAE; as donor quality decreases, CEPAE outperforms them.
Ground-truth metrics: Across all synthetic and semi-synthetic datasets and both intervention settings, CEPAE attains the lowest of MAE, indicating more accurate counterfactual trajectories than the baselines. of MBE further shows that CEPAE substantially mitigates the systematic bias observed for CVAE; among the antuacioonaation-rreeeie
modols, it is best or comparable across datascts. LSTM and AB-LSTM can occasionally achieve slightly better MBE because they are forecasting models (not abduction-actio prediction) and are not biased toward reconstructing the factual post-event series in the same way. For CAAE in the confounded synthetic settings, we report because adversarial training was not stable under our hyperparameter search, and we avoid over-interpreting degenerate counterfactuals.
Ablation and computational cost. To isolate the role of entropy penalization, we train the CEPAE architecture without EP regularization on the unconfounded semi-synthelie dataset. This yields of MAE 0.256 + 0.009 and of MBE -0.250 + 0.008, substantially worse than all baselines and strongly biased, sup- porting that EP is crucial for counterfactual use of the model. We also report wall-clock training times on the unconfounded synthetic dataset (mean over 5 seeds): CEPAE 9:51 + 0:33 (min:s), CAAE 11:31 t 0:41,
CVAE 14:13 + 0:23.
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Table 2: Combined results across datasets for ey=0 and ey=1 (10 seeds). We evaluate both intervention directions: setting 0 where ey=0 and c and setting 1 where 6,=1 and c, Top block: ground- truth metrics (cf MAE/cf MBE; only available on synthetic and semi-syuthetic datasets). Middle: Added Variations /Total/Altored Altered Steps). Bottom: axiomatic metrics (Reconstruction, Reversibility, Effoctiveness).
indicates not applicable (no counterfactual ground truth) or not reported due to unstable training.
Synthetis
Conf. synth.
Semni-yynth
Real world
Metzle
Method
0
1
1
Ground-truth metries (synthetic and semi-synthetic only)
LSTM
+19 .05
.198 005
.199 +
+
.101 004
.80 .002
AB-LSTM
.011 + .06
.197 000
.02 +
.202 = 011
.103 003
083+ .003
of MAE
CVAB
1141 021
.17 014
.115 = 007
.187 = 017
=10== .00
083 + .001
CAAE
.087 1 .012
.083 .011
.065 0004
.061 4 .003
CEPAR
.066 +: .007
.068 + .009
.088 + .300
.086 + .016
.056 + .003
.056 + .004
LSTM
.001 + .011
.001 = .014
-.001 + .010
-0.016 = .016
.003 = .001
.002 + .004
AB-LSTM
.001 + .015
.001 = .011
.001 + 0011
-.001 = .012
.001 = .00
.003 + .011
of MBE
~
CVAE
-67 .019
.067 = 021
.051 + 017
.126 = 028
.011 = 010
-001 = .009
CAAR
.000 + .044
0.018 .021
0000 + .010
-00 + .008
CEPAE
002 007
.077 + .013
.039 + 002
007 + 003
.0000 = .004
.002 + .004
Added Varlations (No counterfactual ground truth required)
Total Steps
CVAR
.457 + .091
.443 -06
.450 + .038
.400 + .00
.037 - -011
.045 + 11
.899 +: .024
1.372 + .070
CAAE
910 .000
.87 006
.510 = 048
.582 + .039
.80 + 100
1.293 + .24
~1
CEPAE
.946 + .04
.931 + .061
.930 + .051
.935 + .048
.747 = .052
.750 +: .048
8419 + 223
1.183 - .098
Altered Stepe
CVAR
388 + 097
360 2 (090
.211 + .188
.246 + 002
.109 = 011
.009 + .015
.312 + 006
.710 121
CAAE
838 1 035
.29 1: 0138
.273 0991
289 + 0073
.445 +: .196
.729 +: 071
CEPAE
.84 +: .010
.833 + .055
.883 + .038
.866 + .033
.300 = .015
.468 +: .010
.558 + 200
.794 + .111
Axlomatic metrics (no counterfactual ground truth required)
CVAB
.1164 .X15
.116 000
.135 + 003
.136 + 0012
.081 = 000
.101 + .X16
.065 .008
J061 MIS
Recoustruction
CAAE
.055 1 .044
.058 .002
.053 = .004
063 1 0003
.038 .008
.044 .077
CEPAE
.051 + .00
.057 = 008
.048 + .004
.048 + .004
.015 = .004
.059 + .004
.099 + .00
.042 + .007
Reversibility
CVAR
127 + XX1
.150 .014
.152 + MI12
.155 = 0005
.100 = .015
.17 = 0016
.U73 .009
.078 + IXI7
CAAE
.66 + .088
.069 :: .008
.063 .005
.067 + .X16
.U59 +: .004
.055 1: .X15
CEPAE
.068 4: .011
.063 + .009
.060 +: .006
.060 +: .005
.050 + .004
.064 +: .005
.052 +: .005
.054 :: .004
CVAE
0.0
1.0 - 0.0
1.0 + 0.0
1.0 + 0.0
.906 = .006
.991 + .001
.631 + .005
.639 + .005
Effretiveness
CAAE
.97 + .X33
.999 0002
(I0S
.995 + .X16
.619 .006
.631 .X16
CEPAE
.999 1: .002
1.0 0.0
.999 1 XXI1
1.0 + 0.0
.IUT
.997 + .003
.627 .005
.621 .006
Proxy metrics: Added Variations and axiomatic metrics. On Added Variations and the axiomatic metrics, CEPAE is closest to the ideal behaviour in almost all settings and typically achieves the best Re- construction and Reversibility while remaining comparable in Effoctiveness. Together, these metries indicate that CEPAF's counterfactual mechanism better respects the desired counterfactual properties, including on real-world data where ground-truth counterfactual errors cannot be computed.
Comparison with Synthetic Control under Controlled Donor Quality Evaluating CEPAE against external benchmarks is challenging. To the best of our knowledge, this is the first work that applies a deep- learning approach based on SCMs a time-series counterfactual problem in the Pearlian sense, so there is no directly comparable baseline in this line of work. At the same time, many methods based on synthetic control exist, but comparing to them poses a major challenge. As synthetic control depends on control time series, its performance depends how predictive they are about the target time series more than in the modeling itsclf. On the other hand, in order to obtain distance metrics with respect to the ground truth counterfactual, we need synthetic data. If we use synthetic data, we should create ourselves the control data, so the measured performance of synthetic control methods would depend basically on how predictive we docide to make the control time scries about the target time series. For this reason, despite being two competing options to solve the same problem, comparing synthetic control with our method is not the best way to evaluate any of them.
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Synthetic Control
0.8
Post-event MAE (mean I 1d)
Causal Impact
CEPAE
0.6
0.4
0.2
0.0
0.1 (R900.911) (R0=000)))))
0.2 (R2=0.976)
0.3 (R210.961)
0.4 (R3
0.5 (R
0.6 (R =0999) 909)
0.7 (R==0.887)
0.8 (R25)
0.9 (R
1.0 (R
Control time series noise std (a)
Figure 4: Synthetic Control and Causal Impact comparison.
Despite this, we have performed an experiment of a synthetic control comparison, where we compare CEPAE with several synthetic control scenarios, each having different level of correlation between control (or donor) and target time series. This experiment is based on the synthetic dataset. We have created, for every "target" time series (i.e., the one over which we want to compute the counterfactual) without observed event (e, = 0), two additional "control" time series or donors (i.e., the time series that we use to predict counterfactual values). Then, we use training data to predict, with an LSTM based model, our target time series from the control time series. This operationalizes the synthetic-contin with strong pre-period fit using an LSTM as the combining function, thus being a strong synthetic control baseline. Additionally, Bayesian structural time series (Almarashi & Khan 2020). In the case of Causal Impact, different model is fit for each target time series, whereas in the LSTM --sed syuthetic-contre baseline a single model is trained to approximate all the target series from their corresponding controls. At test time, we evaluate how well the prediction (performed with test data) of the post-event scenario fits the ground truth counteraactual.
ceincplle-construttie
the target from donors
also compare with Causal Impact (Brodersen et al. 2015), model based on
We create control time series by adding random Gaussian noise with mean 0 and a specific standard deviation (a) to the target time series. With this, we ensure that they are predictive, to some degree, of the target series, being less predictive the o increases. We report also, for every experiment, the R² metric of the predicted pre-event target time series from donors.
We show the results in figure Each experiment has been repeated with 10 different random seeds, and corresponds to the 'Synthetic experiment in table 2 where the MAE for CEPAE is 0.068 0.009. The first experiment (a = 0) corresponds to control time series identical the target one, therefore being absolutely informative. As expected, the error is very low, almost For a = 0.1, a very low degree of noise, the error is slightly lower than CEPAE's error. For 0.2, the error higher.
This experiment allows see why fair general comparison between CEPAE and synthetic control methods can not be established, as results for synthetic control will strongly depend on control data while the SCM based model does not use it. Thus, the selection between our model and synthetic control methods should be problem specific and based on the quality of the control data, in case it is available. In appendix[[ we perform a similar experiment with the semi-synthetic datasct. In this experiment, we observe a similar pattern: when the control time series are highly predictive, the synthetic control method outperforms CEPAE; when they are less predictive, CEPAE outperforms synthetic control.
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6 Conclusion
We have adapted the theory of SCMs and the abcluction-actioo-prediie terfactual estimation. Three autocncoder based models have been proposed: CVAE, well known in coun- terfactual literature although not previously applied to time series problems, CAAE, inspired by image manipulation works, and CEPAE, novel model for counterfactual inference based on an EP. While CVAE shows some superiority with respect to the simple forecast counterfactual estimation in some cases, and CAAE surpasses CVAE in most metrics, CEPAE generally outperforms the other models for most settings, By featuring the reconstruction power of regular AFs instead of VAFs, while not requiring an adversarial training, it makes the interesting option for cases similar to ours. The convenience of CEPAE over synthetic control techniques will depend on the availability and informative capacity control data and the amount of historical event and event-less time scrics.
procedure to time series coun-
cattiin--acttoo--rdcitii process, underexplored in time scries counterfactuals, an
7 Acknowledgments
We gratefully acknowledge Novartis for sponsoring the industrial PhD. We also thank the Government of Catalonia's Industrial PhDs Plan for funding part of this research. We acknowledge Horizon Eu- rope Programme under the A14DEBUNK Project (https://wwe http:/aiddebuun.c.m eu), grant agreement num. 101135757. Additionally, this work has been partially supported by PID2019-107255GB-CCM funded by 100011103 and JDC2022-050313- funded by MCCK//AB/110.303///10000000 by European Union NextCenneatioonUU/PRRR..
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Mattia Prosperi, Yi Guo, Matt Sperrin, James Koopman, Jac Min, Xing Hc, Shannan Rich, Mo Wang, Iain E Buchan, and Jiang Bian. Causal inference and counterfactual prediction in machine learning for actionable healthcare. Nature Machine Intelligence, 2(7):369 (715:,6-777, 2020.
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Paul R. Rosenbaum and Donald B Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41-55) 1983.
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Axel Sauer and Andreas Geiger. Counterfactual generative networks. In 9th International Conference on Learning Representations (ICLR 2021), 2021.
Xinwci Shen, Furui Liu, Hanze Dong, Qing LIAN, Zhitang Chen, and Tong Zhang. Disentangled generative causal representation learning, 2021. URL
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DJ Strouse and David J Schwab. The deterministic information bottleneck. Neural computation, 29()): 1611-1630, 2017.
Naftali Tishby, Fernando C. Pereira, and William Bialek. The information bottleneck method, 2000. URL
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Anwaar Ulhaq, Naveed Akhtar, and Ganna Pogrebna. Efficient diffusion models for vision: A survey. arXiv preprint arXiv:2210. 09292, 2022.
Arnaud Van Looveren and Janis Klaise. Interpretable counterfactual explanations guided by prototypes. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 650-665. Springer, 2021.
Athanasios Vlontzos, Bernhard Kainz, and Ciarán M. Gilligan-Lee. Estimating the probabilities of causation via deep monotonic twin networks. CoRR, abs/2109.01904, 2021. URL httpp://arxii.orr/aabb///
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A Industrial Application
In many industries, knowing the impact of competitor entry sales is critical for business planning, investment allocation and objectives setting. The pharmaceutical industry is notably affected when drug's patent expires and Loss of Exclusivity (LOFF) (Castanheira et al.
2019) takes place, prompting competitors to launch cheaper generic versions of the drug. This usually results in a dramatic decrease in the sales volume of about 60-70% in the first years (Castanheira et al. 2019), severely affecting company revenues. Thus, accurate assessment of the market impact of generic drug entries is of utmost importance.
Time series counterfactual estimation is an essential tool to understand the impact of an event time-serics data. It is applied to situations where event or a trealment at certain point in time alters time series' trajectory, and consists in inferring, once given the observed post-event data, the counterfactual data, i.e., the time series that would have taken place if the event had not occurred
In the case of an event such as the entry in the market of a generic competitor drug, a straightforward approach for counterfactual estimation to simply use a time series forecasting model trained on historical data which was unaffected by this type of event, and predict the next steps of our pro-event time series according to the model. The main problem of this approach is that its estimations rely solely on pre-event data, lacking the ability to incorporate relevant information of the post-event facts that could have affected the counterfactual time series.
For example, let us imagine that few months after a generic drug enters the market to compete with our target brand, an extreme weather event creates a problem logistics which results in lower than expected sales in the whole pharmaceutical market. With a forecast-based model we would estimate regular, not reduced, counterfact ual sales volume despite the fact that, in all probability, our counterfactual time series would have been affected by the aforementioned happening. This would create misconception of the impact that would be problematic for business analysis and planning.
In our company, once we have sales observations for a certain number of months, we are interested in knowing what the impact LOE has been. For that, we pass the sequence of post-event observed values to our model, compute the counterfactual, and then obtain the impacts as the difference between the observations and the counterfactuals. This provides useful information for business planning, and allows to obtain information of impacts that can be used to predict future impacts.
B Extended Related Work
Several mothods have been proposed for time series counterfactual estimation and causal impact inference in the presence of an event treatment. Difference in differences (Callaway et al., 2024) is a common approach which requires some control time series that are not affected by the event to estimate counterfactuals. This method features some important limitations like the Parallel Trends Assumption. Synthetic control (Bouttell ct al. 2018 Abadie & Gardeazabal 2003), which generalizes Difference in differences and overcomes some of its limitations, selects several available time series, other than the target one, that have not been affected by the event, computes some weights based on their pre-event similarity to the target time series, and then estimates the counterfactual as the weighted average of the post-cvent control time series. Causal Impact [Boodersen et al.] 2015) is closely related to the synthetic control approach, and its main difference is that it estimates counterfactuals through a model that predicts the target from control series trained with pro-event observations. Matrix Completion Methods (Athey et al.] 2021 are an alternative that can be viewed as a combination of synthetic control and the more classical unconfoundedness approach (Rosenbaum & Rubin
1983 Imbens & Rubin 2015
In recent years, many works have appeared that mix the structural causal model (SCM) theory (Peall 2000 and deep learning techniques to estimate counterfactuals. Apart from the ones mentioned in the paper, based on VAË's (Kingma & Welling 2013) and, to lesser extent, normalizing flows (Kobyzev et al. 2020 other
interesting approaches have been proposed. For example, Shen et al. (2021) use GANs (Goodfellow et al.
Note that it is also possible for the opposite scenario: the observation might he when the event doesn't occur, and the conntrfacttial when it does.
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2014 while Jeanneret et al. (2022) uses diffusion models (Ulhaq et al. 2022 Montciro al. 2023 presents some useful metrics to evaluate counterfactuals, which are used in our paper to evaluate the models. Sauee & Geiger (2021) uses deep neural networks to disentangle object shape, object texture and background in natural images. Van Looveren & Klaise (2021) utilizes class prototypes in order to find interprotable counterfactual explanations. Parascandolo el al. (2018) uses multiple competing models in order to retrieve a set of independent mechanisms from a of transformed data points an unsupervisod way.
It is important to notice that there several works in the Explainability field that use the term counterfac- tual in completely different sense than this work. In the Explainability context, if we consider, for example, binary classifier and a given input, the term counterfactual refers to the most similar input to the one given that delivers a different classifier outcome. This approach can help understand how the classifier works, but is different from the causal concept of counterfactual. Some works that tackle the Explainability counterfac- tual problem are applied to time series (Wang et al. 2023 2021 However, it important to recognize the difference between this approach and the causal counterfactual problem that our work addresses.
There other interesting works about causal representation learning with deep generative models that do not tackle directly the problem of counterfactual inference but are interesting take into account. [ocaoglu ct al. (018)) and Liu et al. (2019) combine GANs (Goodfellow et al. 2014) with SCMs, basing a generator architecture on an assumed causal graph. However, this method lack tractable abduction capabilities and
therefore cannot generate counterfactuals, reaching only the second rung of the causal ladder (Pearl 2000).
Yang et al. 2022 proposes a method that learns a causal model, including the directed acyelic graph (DAG), over latent variables from data, and generates counterfactual samples. Kumar et al. (2023) uses GAN based approach to address the specific problem of spurious correlations in medical datasets.
Finally, there are other interesting counterfactual estimation approaches that are applied to tabular data. Among them, Yoon et al. 20188), based on GANs, and Vlontzos et al. (2021). based on Deep Twin Networks, similar to Siamese networks Koch et al. 2015
stand out.
C Extended Information Theory Background
The formula for the differential entropy S(X) of continuous random variable with probability density function f(z) over its support is given by:
S(X) - JJ- f(x) log f(x) dr.
(13)
The conditional differential entropy S(XIY) of continuous random variable X given another continuous random variable Y with joint probability density function fx, y) and conditional probability density function (xly) is given by:
J. Ix - v) log (xly) dr dy
S(XIM)=
(14)
The joint entropy S(X, Y) of two random variables X and Y with joint probability density function fx,y (z, v) is given by:
s(x, = x-x,)=--(x fx, u) log fx,y (2, y) do dy
(15)
The conditional joint entropy S(X, Y|Z) of two random variables X and Y given a third random variable Z with joint probability density function fx.y.z and conditional joint probability density function fx.y 2(x y/z) is given by:
S(X,Y12)= = J (x, log fx,xiz(a, v1z) de dy
(16)
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The mutual information I(X; Y) between two contimuous random variables X and Y with joint probability density function fx, v) and marginal probability density functions fx (x) and Jr(y) is given by:
I(XY)) = -x fx
fx,((z))) dr drdy fx(x)fr(y)
(17)
The mutual information can have values from 0 SX) when and are the same distribution.
The conditional mutual information I(X; 2| between two continuous random variables X and Y given a third continuous random variable Z with joint probability density function fx, y. and conditional probability density functions fx,y] vl=), (x|z((|22) and fr|z(ul=) is given by:
I(X fa. J,, fx.,2(0,,))) log
fx.y z(z,8lz) fxiz(ala)fy z(yl)) dx dy dz
(18)
All the previous relations are valid also for discrete variables if integrals are changed by summations.
For discrete variables, the entropy is interpreted as measure of how informative, and uncertain, a random variable is, or as the mean amount of information that its outputs bring (in the case of conditioned entropy, how informative a variable is once conditioned on another variable), and the mutual information is interpreted as the amount of information that one variable brings about another one. Sometimes, the differential entropy is interpreted in the same way as the discrete entropy and, in this work, do in an intuitive level. However, there are some limitations with respect to this. For example, the differential entropy can have values lower than 0, and change of scale can modify it, which is counter intuitive. Nevertheless, the mutual information preserves its mcaning for two continuous one continuous and one discrete variable, which is fundamental for our methodology.
the main paper, we state these four information theory equations that are essential for the development of CЕРАΕ:
S(X) +Xx;))
(19)
S(X,Y) =
+ + I(x;Y),
(20)
S(X,Y)= ((X) ((1)X),
(21)
S(X, S(Y12))
(22)
Relation 21 is the chain rule of entropy, which is an established result in information theory, and relation 22 is the chain rule of entropy in the context of conditional entropy.
Relation is simply a rearrangement of the definition of mutual information:
I(X;Y) = S(X) S(X(r).
(23)
To demonstrate relation
20 using the chain rule of entropy, we have:
Y(X,Y) -())))
(24)
From the definition of mutual information:
:X; S(((x)
(25)
Rearranging this equation, we find:
S(YI)) S(Y) - I(X;Y).
(26)
Substituting this back into the chain rule expression for joint entropy, we obtain:
S(X.Y) ((X) -(;Y).
(27)
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C.1 Entropy Maximization by the Gaussian Distribution
Theorem: With normal distribution, differential entropy is maximized for a given variance. A Gaussian random variable has the largest entropy amongst all random variables of equal variance, alternatively, the maximum entropy distribution under constraints of mean and variance is the Gaussian (Cover) 1999..
Proof: Let g(x) denote a Gaussian probability density function (PDF) with mean μ and variance o², and let f(x) be an arbitrary PDF with the same variance. Since differential entropy is invariant under translations, we can assume without loss of generality that (() shares the same mean p as g(x).
The Beelback-Teibbler divergence between these two distributions is given by
0 < DxI((1l9)) = /- f(x) log
[(x)) ((x))
dr = -S() - L ((x) (ox)log((e dx.
Next, consider the integral involving g(z)):
(e-m)² 20d
f(x) logg(r)) de f(x) log
dx.
V2TT²2
This can be expanded and simplified as follows:
[ f(x) log ((x) da
-
(x---)) u1)
f(x) (o))oog V2T0² dr + f(x)
dx
20
log((TTo²) - log(e)
1
log(2Treo))
Sggg).
with equality holding if and only if f(x) = g(x), as dictated by the properties of the Kullback Leibler divergence.
CVAE Description
The loss function for a classic CVAE (Kngma & Welling 2013) with a 3 penalty (Higgins et al. 2017) which correspouds to the evidence lower bound (ELBO), for a datum x and a conditioning c, is given by:
LOVAR(O, Φ) = Ex po(z|z, c)] - 3 K KLggo c) p(=))]
(28)
where both go(2|z, e) and co( are set of dimension-wise independent normal distributions parame- terised, respectively, by an encoder neural network Es and a decoder neural network Dg, p(z) is an isotropic divergence (Hall 1987) and / is a penalization over KL
normal prior distribution, KL is the Kullback-Leibe (Higgins et al.] 2017). In the model training, the ELBO function maximized with respect to parameters of the neural networks using the re-paramctrization trick to sample from the approximate latent postcrior: - == c) + (xs c) O2 I).
Based on the time series setting and the encoder-decoder counterfactual estimation method described in the main paper, it is possible to generate a counterfactual sample def by encoding an observation 9, and its parents h and é,, i.e. obtaining the normal distribution go(2lur, h, ey), where position and scale parameters come from the encoder: μe. as = Es(y,, hy),,, then sampling the latent posterior from this distribution: z ~ (z(z)yy h, ey), and finally decoding along with the counterfactual event eef² des ~ po h, eef). Notice that, at a practical level, the counterfactual will be decoded from the latent sample in a deterministic way: jef Do (2, h e.f).
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E CAAE Description
For a data instance (x', c²), let (E c')) be a trainable classifier or regressive estimator of C (depending on whether C is categorical or continuous) with parameters v,, then we need to use a conditioner loss Coa 4b) Co'aacc, (Fs c))); Φ, 4), which can take different expressions depending on that kind
of conditioner or conditioners we have. In an adversarial game, Gy has to maximize this loss and Eg has to minimize it. Considering that the reconstruction loss for a conditioned regular AE is:
(0, = =1|=²= -Do(EDDo
(29)
the total loss of CAAE is:
CAAE Φ, 2/) = (0, - (on
(30)
where controls the trade-off between the quality of the reconstruction and the invariance of the latent representation. To implement the adversarial training using backpropagation, we use the Gradient Reversal
Layer (GRL) (Ganin al.] 2016)) and, as Lample el. al. (2017) slart off with a À value of and increase it linearly for each iteration.
30 should reach saddle point (0, c 3) that achieves the equilibrium between invariance of representation and reconstruction:
By using the objective in eq-
( arg min CCAAE (b)
0,6
(31)
$ arg max CCAAE(0, 2.
&
F Further Discussion on Equation 9 terms
In a continuous and deterministic setting (CEPAE is a deterministic model), the conditional distribution p(Z C,X) collapses onto (near) Dirac delta i.e., Z is one-to-one function of (C, As result, the conditional differential entropy S(Z C)X) tends to co.. Simultancously, the mutual information I(X;2, C) can diverge to +o0, because knowing (Z, C) almost determines X. Despite these individual divergences, their sum in Equation 9 must remain finite as long as S(Z) itself is finite - intuitively, the large negative and large positive components cancel each other out in the equation, because the rest of the terms are all finite. Morcover, S(Z) stays finite unless p(Z) degenerates entirely This is exactly what, would happen if the entropy penally was the only loss, but the reconstruction loss in our framework prevents complete collapse, ensuring that Z retains some spread and thus retains a finite marginal entropy S(Z). Therefore, while individual terms such as I(X; Z, C) and S(Z C, may formally blow up under deterministic mappings, their sum remains definee and finite, which preserves the overall validity and meaning of Equation 9.
a delta distribution.
Beyond this, in practice, Z stored and processed in finite-peccsioon floating-poin format. This means actually takes values from large but finite set of representable numbers, effectively making it discrete. In a discrete setting, mutual information cannot diverge to too: there is always an upper bound determined by the (finite) cardinality of the variable's domain, we do not observe true infinite negativo-infiite entropies. Likewise, conditional entropies in a purely discrete model are nonnegative, ruling out the -oo scenario. Thus, although we might treat Z as continuous in theory, real implementations quantize Z sufficiently to keep the relevant information measures finite.
G Information Bottleneck
The Information Bottleneck (IB) framework (Tishyy et al. 2000 is an informatiion-heorrliee, method for learning latent representations. Given an input source X E X and a corresponding output target Y e y, the goal to learn an encoder Pa (Z | X) that captures only the information in X that is relevant for predicting Y. Formally, IB seeks to find sufficient statistics of X with respect to using as little information from possible. The standard IB objective is expressed
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(1x:22)-31X::22)
min (2
(32)
where 3 > O is a hyper-parameter balancing how much information from X is retained in Z.
Originally, the TB problem was solved with methods like Tterative Blahut-Arimote algorithms (Yeung & Yeung) 2008), deriving self-consistent equations for the conditional distribution Po(Z X) (and related marginal distributions) using variational calculus and Lagrange multipliers. This iterative method works well when X, Z, and Y are discrete and the state spaces are relatively small. However, scaling these methods to hign-dimensional or continuous domains is challenging. On the other hand, (Alemi et al., 2017) presented a method to achieve the IB objective with neural networks based on a variational approximation. In fact, VAEs could be seen, with some reservations, as an application of this method, although the original development of VAFs (Kingma & Welling 2013) does not mention this relationship. number of works have successfully applied IB for several purposes, like reducing generalization error.
There are important differences between this method and ours. First, IB tries to minimize mutual information among inputs and representations, while our method aims at minimizing the entropy of the representation. Also, our model is deterministic, avoiding the imprecision problems that probabilistic frameworks can show. An idea that is more similar to our method is the one presented in (Strouse & Schwab, 2017). This work proposes a different objective which aims at reducing the entropy of the internal representations instead of the mutual information between the internal representation and the input. The objective can be expressod as:
min (S(Z)-BI(Y:22)) Pr (2
(33)
If we consider that the objective Y is the same as the input X, as in our autoencoder, this objective corresponds, in an information-theorelie sense, to CEPAF's objective. However, there are very substantial differences between this work and ours. First of all, (Strouse & Schwab 2017) proposes to reach their objective through algorithms similar to those originally used in the IB framework, i.c., they do not employ neural network techniques and therefore the method has important limitations regarding applications to continuous domains. Thus, our implementation of the entropy penalty, and its integration into a neural network model, is distinct from their approach. On the other hand, (Strouse & Schwab) 2017) always consider Y to be different from X and do not mention the possibility of using their framework in an autoencoder architecture.
To the best of our knowledge, no prior approach has explicitly used IB or a similar information--hheore strategy to disentangle a latent variable from a conditioner estimate counterfactuals.
H Identifiability Limitations in Multidimensional Data
Many works on counterfactual estimation overlook identifiability, mainly because it is usually assumed that, in markovian graphs, all counterfactual queries are identifiable. However, recent works like Nasr-Esfahany & Kiciman 2023) affirm that markovianity alone is insufficient and present more restrictions. In concrete, that paper provides an impossibility result for identifiability of generation mechanisms with multidimensional exogenous variables, which affects all counterfactual works applied to multidimensional data 3 including ours. Nonetheless, as Nast-Esfahany & Kiciman (2023) notes, "cxact counterfactual identifiability is often too strong, e-g.; in cases where low counterfactual error is tolerable by practitioners". Otherwise, all works on counterfactuals in multidimensional data, like Pawlowski et al. (2020) or [Sanchez & Tsaftaris (2022). should be invalidated. In table 1 of the main paper, we show that our method achieves very reasonable counterfactual errors, and deciding if they are tolerable will depend on each application.
Fvee if one dimensional latent variable assumed, that does not lead identifiabilitty in multidimensional settings Nasr- Fhar-Fefahany al. 2023).
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Metrics from the Axiomatic Definition of Counterfactual
As mentioned in paper, Montciro et al. (0223) proposes three metrics to mcasure soundness of coun- terfactual inference model without having access to ground truth counterfactuals. It is rooted in the Judea Pearl definition of counterfactual (Pearl 2000), the soundness theorem (Galles & Pearl 1998p, and the com- pleteness theorem (Halpern 2000), which, together, state that composition, effectiveness and reversibility are necessary and sufficient properties of counterfactuals in any causal model. Let x be an observation with counterfactual parents pa, and x² a. counterfactual of with parents pa". Then, counterfactual function can be defined in such a way that x²' == f(z, pa, pa"), where the abduction of the exogenous noise e is implicit. With this notation, where there is distinction among the ideal counterfactual function f and its approximation with counterfactual model f. Monteiro et al. (2023) defines the axioms that an ideal coun- terfactual function must obey and propose, in relation to each axiom, metric to evaluate approximated counterfactual functions. The three metrics are the next ones:
Intervening on variable to have the value it would otherwise have without the intervention will not affect other variables in the system. This implies the existence of a null transformation f(r pa, pa) = x since pa" = pa, then æ is not affected. Since the ideal model cannot change an observation under the null transformation, we can measure how much the approximate model deviates from the ideal one by calculating the distance between the original observation and the mth time null- till--raanformed observation. Given a distance metric dx, such as Mean Absolute Error (MAE) (which has been selected in our work), an observation x with parcnts pa and functional power (which is always 1 in our work), can mcasure composition as Composition == dx (e, pa. pa)).
(1) Composition:
(2) Reversibility: Reversibility prevents the existence of multiple solutions due to feedback loops. If mechanism is invertible, this means that if x² == f(x, pa, pa"), then f(x", f(=",pa", pa). In other words, the mapping between the observation and the counterfactual is deterministic for invertible mecha- nisms. For a further discussion on this topic, see Monteiro et al. (2023) Thus, it is possible to measure reversibility by calculating the distance between the original observation and the cycled- back transformed observation. Setting p(m) (zr, pa, pa*) == i (M(x; pa, pa"), pa", .pa. given distance metric dx, an observa- tion with parents pa and functional power (which is in our work), we measure reversibility Reversibility(m) (z. pa. pa") == dx (z,, (m)) (zz, pa, pa)). The chosen distance metric in our work is MAE,
variable to have a specific value will cause the variable to take that value. Thus, suppose Pa is an oracle function that returns the parents of a variable, then we have the following equality: Pa((f, pa, p)))). Effectiveness is difficult to measure objectively without relying on data- driven methods. Following Monteiro et al. (2023), we measure effectiveness individually for each parent by creating a pscudo-oracle function Pag, which returns the value of the parent pag given the observation. Using an appropriate distance metric dg, such as accuracy for discrete variables or 11 distance for continuous ones, päg, (aas)aaa))))))))))) pat).
(3) Effectiveness:
Intervening
measure effectiveness for each parent as Effectiveness, pa, (a,ppa)) di. (Pak (f,,
We have given general definitions these metrics, using generic notation. In our work, the observations correspond to post-event time series, the parents pa correspond to the factual event and the pre-event time series, and the counterfactual parents pa" correspond the counterfactual event and the pre-event time scries.
J Added Variations Metrics
The added variations metrics are implemented as follows: for each time series to be evaluated, several positive and negative values in the order the data values are chosen; for each of this values, several windows of few consocutive steps from the post-event time series are selected and the chosen value addod to those steps. After that, a counterfactual estimate is obtained for every altered time series and it is compared to the counterfactual estimate of the non-altered time series. Two quantities are obtained: 1)
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Total difference, that takes into account the difference among the altered counterfactual and the base counterfactual in all the steps, and 2) Altered steps difference, which takes into account the difference among the altered counterfactual and the base counterfactual only in the steps affected by the alteration. Altered steps difference is the metric reported in the main paper. These quantitics are then divided by the expected difference (the product of the alteration value and the number of affected steps). Thus, ideally, the final results for both total difference and altered steps difference metrics should be 1. Next, we give a formal definition of these metrics.
Let y (y. E T be the post-event time series over which want to perform counterfactuals, h its corresponding historical time series previous to the event, ef the (factual) event, and def fly. h, h, Cef). where i is a counterfactual function and eef is the counterfactual event, its corresponding counterfactual estimation. Then, consider time series A = 0...0, U 0...0 with T steps, where VA is the value of the altcration which is added only to a certain number of consccutive steps. Let y = y + A be the altered time series, then D. would be ils corresponding counterfactual estimation. We consider that, if our counterfactual model is correct, alterations in the factual time series should be reflected in the counterfactual time series. Thus, ideally >, E2 - }cf()) = Σ; A; = n UA, where n is the number steps affected by the alteration in A. Taking into account that we use different time series A with different values n, and VA. we can express total differences metric for single time series y (TD) as:
Fe(() ())(s)
TD
(34)
Tha VA
and altered step differences (ASD) as
Σx (e(()) na UA
Def(
ASD
LiESA
(35)
where S is the set of altered steps (those with value da and not 0) in A and II is the indicator function. We see that, ideally, the result of these averages over the different alteration schemes should be 1. The results given in the paper are the averages of these metrics over all the time series in the test set. The parameters nA, S and a are particular for every dataset and can be seen in the code.
K Adversarially Balanced LSTM
This appendix details the AB-LSTM baseline referenced in Sec. 5.3 of the main paper. The model adapts the udversarial-balancee idea of the Counterfactual Recurrent Network (CRN) (Bica et al., 2020) to the single, one shot event setting studied here.
CRN learns a latent summary that is predictive of future outcomes yet independent of treatment assignment. We transfer this principle to our scenario, where the treatment/event variable E (0, 1} occurs once at time To.
Let the pre-event history be (#i...... ......,,.. ,T,,) ie. a realisation of the random variable defined in Sec. 4.1, and let denote the binary event flag. AB-LSTM is composed of:
LSTM network dy with hidden D; it maps + 2 E E...
2. Outcome head fo: a two-layer MLP that predicts the next T" steps, &T.+1ITTT = fz(zz)).
Event discriminalor gø: a logistic classifier that tries to recover e from z.
A gradient-rewersal layer (GRL) Ganin et al. 2016 with coefficient À is placed between a and IV, so gradients flowing to the LSTM network are multiplied by -X, forcing to obscure e.
Training objective. For sample (h, e, v) we solve
min max [5((,,)) vlli + X (C),) adversarial balance
2 = (N))
0,0 &
prediction MAE
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Published in Transactions on Machine Learning Research (obbuaayy/2200
The weight À is ramped linearly from to Amaz over the first 40 % of updates (Ganin et al. 2016
Counterfactual generation. At test time, given factual pair ehy,
Dr = fo(oo(hs), ef). der - (r-(dbo(bhs)... 1-
Because no post-event data are required, AB-LSTM acts as a forecasl-only counterfactual estimator, in the same way as the LSTM baseline. Both of them receive the inputs e), and do not consume post-event values.
L Synthetic Control Experiment in Semi-Synthetic Dataset
We perform an experiment similar to the one in the main text with the synthetic dataset. This experiment is based on the semi-syuthetic dataset. We have created, for every "target" time series (i.e., the one over which want to compute the counterfactual) without observed event (ej 0), two additional "control" time scries (i.e., the time series that use to prodict counterfactual valuos). Then, predict, with an LSTM based model, our targel. time series from the control time series. At test lime, we evaluate how well the prediction (performed with test data) of the post event scenario fits the ground truth counterfactual.
training data to
We create control time series by adding a random gaussian noise with mean and a specific standard deviation (a) the target time series. With this, we ensure that they are predictive, to some degree, of the target scries, being less predictive the increases.
Table 3: MAE of the synthetic-control baseline for different noise standard deviations a.
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
We obtain the results in table 3 Each experiment has been repeated with 10 different random seeds, and corresponds to the semi-synthetic experiment in table from the main paper, where the MAE for CEPAE is 0.056 + 0.004. The first experiment (a 0) corresponds to control time series identical to the target one, therefore being absolutely informative. As expected, the error is very low, almost 0. For 0.1, very low degree of noise, the error is slightly lower than CEPAE's error. For 2 0.3, the error is higher.
M Convergence Behaviour Comparison between CAAE and CEPAE
In this section, we compare the convergence behaviour of CEPAE and CAAE. The experiments are based on the unconfounded semi-synthetic dataset. We monitor how the counterfactual mean absolute error (MAE). computed using ground-truth counterfactuals on a held-out set, evolves during training. In Figures 5 7
and 9 we report this counterfactual MAE every 5 epochs, from epoch to 350, under different training regimes.
Figure shows the standard training regime for CEPAE, with the EP weight fixed to 0.09 during the whole training. The model converges slowly but steadily and reaches fairly stable region around epoch 100. Figure shows the behaviour when we add lincar schedule for the EP weight, analogous to the one used in the main experiments: the weight increases linearly from 0 to 0.09 during the first iterations. In this case, CEPAE converges faster, reaching stable values around epoch 50, with a slight further decrease of the loss afterwards.
Figure 7 shows the behaviour of CAAE when its adversarial weight follows a linear schedule that increases it from to its final value 8.9. The model quickly attains low MAF around epoch 50, but afterwards the
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Counterfactual MAE vs Epoch
0.24
0.22
0.20
0.16
CF MAE
0.14
0.12
0.10
0.08
0.04
50
100
150
200
250
300
350
Epoch
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.
loss grows slowly yet persistently. In figure S we show a failure mode that we have occasionally observed for CAAE: some epoch, the counterfactual MAE suddenly increases and then remains high for the rest of the training. This an example of adversarial instability.
Finally, Figure iillustrates FigureGillustraees what happens when do not apply the linear schedule to the adversarial weight in CAAE and instead keep it constant throughout training: the model is unable to reach low-loss regions. In contrast, CEPAE with fixed EP weight converges more slowly than with a schedule but still reaches good performance after some additional epochs and, importantly, does so without the abrupt instabilities observed for CAAE. This is particularly relevant real-world applications, where counterfactual ground truths are not available and is therefore impossible select the best epoch via early stopping on a validation set based counterfactual error. The stable convergence of CEPAE makes it more reliable in such settings.
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Published in Transactions on Machine Learning Research (February/200)
Counterfactual MAE Epoch
0.24
0.18
0.16
CF MAE
0.14
0.12
0.06
0.04
250
300
Epoch
Figure Convergence of CEPAE with a linear schedule for the EP weight on the unconfounded semi- synthetic datasct. The EP weight increases lincarly from O to 0.09 during the initial epochs. Counterfactual MAE decreases more rapidly than the weigh- case and stabilizes around epoch 50.
Counterfactual MAE Epoch
0.24
0.18
0.16
CF MAE
0.14
0.12
0.10
0.06
0.04
100
150
200
250
300
350
Epoch
Figure 7: Convergence of CAAE with 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 MAF around epoch 50, the loss slowly increases again, illustrating the instability inherited from adversarial training.
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Published in Transactions on Machine Learning Research (February/200)
Counterfactual MAE Epoch
0.24
0.18
0.16
CF MAE
0.14
0.12
0.06
0.04
100
150
200
250
300
350
Epoch
Figure 8: Example of a failure mode for CAAE under a linear adversarial-weighh schedule. At some point during training, the counterfactual MAE suddenly increases and remains high for the rest the epochs, showing a typical instance of adversarial instability.
Counterfactual MAE Epoch
0,24
0.18
0.16
CF MAE
0.14
0.12
0.06
0.04
100
250
300
Epoch
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.
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Published in Transactions on Machine Learning Research (February/200)
N CEPAE Hyperparameter Sensitivity Analysis
In this section, we analysc the scnsitivity of CEPAE to its two main hyperparameters: the latent dimension- ality and the entropy-penalty (EP) weight. In the experiments reported in the main paper, we use a latent dimensionality of 8 and an EP weight of 0.09. These values were selected after a simple hyperparameter search using the factual data of the unconfounded semi-syuthetic dataset.
Here, we study how the counterfactual MAE changes when we vary these hyperparameters. First, we change the latent dimensionality D= while keeping the EP weight fixed to 0.09. Second, we vary the EP weight AEP while fixing the latent dimensionality to D2 8. In both cases train CEPAE with the same architecture and optimisation settings as in the main experiments and evaluate counterfactual MAE and MBE using the same protocol.
The results are summarised in Tables and
Table 4: Sensitivity CEPAE to the latent dimensionality D, on the unconfounded semi-synthetic dataset. We report counterfactual MAE and MBE (mean one standard deviation across seeds). The EP weight is fixed to 0.09.
Metric
Latent dimensionality D=
6
7
8
10
11
12
MAE
0.062 = 0.004
0.058 + 0.003
0.056 + 0.004
0.051 0.003
0.049 = 0.004
0.055 + 0.004
0.051 + 0.004
MBE
-0.008 + 0.002
0.002 + 0.003
0.002 + 0.004 0.006 + 0.004
0.010 + 0.003 0.006 + 0.002 -0.003 +: 0.002
Table 5: Sensitivity of CEPAE to the EP weight ÅEP the unconfounded semi-syuthetic dataset. We report counterfactual MAE and MBE (mean + one standard deviation across seeds). The latent dimensionality is fixed to D2 = 8.
Metric
EP weight AgP
0.05
0.07
0.09
0.11
0.13
0.15
MAE
0.055 + 0.004
0.051 + 0.003
0.056 + 0.004
0.056 + 0.003
0.060 + 0.004 0.075 + 0.004
MBE
0.011 0.004 0.004 + 0.003 0.003 + 0.004 0.004 0.004 0.003 + 0.002 0.003 0.002
Overall, we observe that CEPAE is relatively robust to moderate changes in both hyperparameters. For the latent dimensionality, MAE varies smoothly across the range Dg E [6, 12]. with no abrupt degradation in performance and MBE remaining close to zero. Similarly, EP weights in the range Agp E [0.07, 0.11] yield very similar MAE and MBE values, while too weak regularisation (AęP 0.05) or too strong regularisation (ΛEP 0.15) leads mild deterioration in MAE. These results suggest that the configuration used in the main text (D= 8, dEP 0.09) lies in fairly flat region of the hyperparameter space, and that CEPAE does not require extremely precise hyperparameter tuning to obtain good counterfactual performance.
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Reviews:
[{"review_id": "qjM37NWPxl", "reviewer": "Reviewer_CdHW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose CEPAE, a novel framework that uses entropy regularization in latent space towards improving counterfactual inference for time series.\n\nThey compare their approach to two established models (CVAE and CAAE) across a range of synthetic, semi-synthetic and a real-world dataset(s). Given the considered metrics, CEPAE exhibits favourable performance w.r.t. competing approaches in the majority of tasks.\n\nWhile the empirical evidence suggests an advantage of the extensions that the authors propose, these are often presented in a way that makes it hard to discern *why* these extensions offer a benefit and how they relate to prior work.", "claims_and_evidence_yn": "No", "claims_explanation": "My main concern with this manuscript lies in the presentation.\n\nAs a general comment, I feel that the contents could be compressed which would also greatly help the clarity of the message the authors wish to convey. Many passages are written almost colloquially, leading to a lack of precision and lack of clear thread. In other sections the number of commas used disrupts the flow of reading. The citation style is inconsistent, with inline citations often being used where standard citations should be used. While these are style issues that can be immediately improved, there are many instances when the authors argues in hand wavy fashion, leading to doubts about the authors claims.\n\nSpecifically, when discussing related works, the authors allude to a specific connection or result with explicitly stating what they mean. This happens throughout, but to single out an example in Section 4.2. they state \"[f]or the problem of counterfactual estimation in high dimensional settings, autoencoder based approaches are the most extended and effective.\" without backing up this claim with any reference. This continues in the next paragraph, claiming that $\\mathbf{Z}$ is equivalent to $U_y$ and immediately stating that the noise is not identifiable. This apparent contradiction is never explicitly resolved. The next paragraph about the importance of disentangled representations is also hand wavy and not backed up by enough precise theoretical statements.\n\nTo repeat: the empirical results suggest that the authors' contributions are useful and the arguments the authors do provide don't seem completely off, but the presentation is not up to par for a scientific paper.", "audience_yn": "Yes", "audience_explanation": "There is a large body of work on counterfactual inference, as well as a large community in time series causal inference so there should be interest from both communities in this work.", "requested_changes": "The paper should be rewritten to be more concise and accurate in the stated claims. If claims can be made in a theoretically sound manner, they should be done this way and if not, they should not be alluded to in hand wavy, speculative fashion, but rather not made. The empirical evidence suggests that the authors are on to something, but the presentation is lacking for publication in this state.", "broader_impact_concerns": "-", "additional_comments": "The reference for the VAE is incorrect. It was published in 2013. Please use the non-arxiv version.", "extra_scores": {}, "extra_text": {}}, {"review_id": "THYxob0yRg", "reviewer": "Reviewer_WBjb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- The authors formalize the problem of estimating counterfactuals in time series.\n- They propose a new type of autoencoder with a penalty that minimizes an upper bound on the entropy of its latent space which should encourage a disentanglement beneficial for reconstructing the counterfactuals.\n- They evaluate their method, along with several existing methods, on a set of experiments.", "claims_and_evidence_yn": "No", "claims_explanation": "Looking at the two contributions from the authors, I've some doubts about whether they provide sufficient evidence to support their claims around these two contributions; below I've tried to summarise my main concerns.\n\n## Contribution 1: adapting the abduction-action-prediction procedure to a time series setting.\nWhile I'm not familiar enough with the literature to assert that the authors are not the first to do this, I'm not fully convinced by how they handle the time aspect of the problem, or how their setup differs meaningfully from a non–time-series setting. If I inspect the graphs in Figure 2, there is no time-series structure in it. I would expect something more dynamic, such as nodes in H having arrows pointing to themselves, which could lead to temporal dependencies not currently depicted.  It also seems like a strong assumption that the event always occurs at the same time point or that the time series has the same length always. This issue is also reflected in the model itself, since I don't understand whether the autoencoders take the temporal nature into account (for instance, it would have felt more natural to use some sort of autoregressive model instead). In the end this concern comes down to one thing: could I not simply think of $H$ and $Y$ as multivariate random variables and ignore the time structure, or is there more to it?\n\n## Contribution 2: Demonstrating the validity of CEPAE both theoretically and experimentally\n- **For the theoretical part**: What are now actually the theoretical claims in the paper? One key claim by the authors is that we need an disentangled latent representation, but it would be helpful then to see some more arguments (or references to prior work in e.g. the non-time series setting) for why this would solve the problem. And you do not provide theoretical guarantees, I would have hoped for more empirical evidence to show that this disentanglement is necessary. A smaller point, but one that feels glossed over as well. You mention that disentanglement is enabled by minimising the entropy, but ultimately the method minimises an upper bound on the entropy. Under what conditions is minimising this upper bound effective? Are there scenarios where the procedure might fail?\n- **For the experimental part**: I'm mainly missing the ablation study which relates to the claim that the disentanglement is critical for doing the counterfactual estimation well, but I've also some other major remarks about your analysis of the experiments which I'm sharing under \"Requested changes\".", "audience_yn": "Yes", "audience_explanation": "I believe the problem setting is of interest to researcher in the field, and especially counterfactual estimation (in the Pearlian sense) have been less studied.", "requested_changes": "## Critical adjustment (to secure recommendation for acceptance)\n- I would like the authors to establish whether the observed improvements of CEPAE over, e.g., CVAE come from the entropy penalty rather than from the reconstruction loss (which is the main argument for not using CVAE). I believe this would require another experiment where you remove the entropy penalty and compare against your current implementation.\n- It is not always clear to me what question you are trying to answer with the metrics used in your experiments, and what reasoning led you to your conclusions. I would hence want a bit more elaborate discussion around the results in Table 1 (more than a single paragraph), because the results seem more nuanced than that CEPAE simply outperforms the other methods. For example, the table shows many interesting trends, yet you state in the results section that “MAE is the most important one,” which raises the question: what is the purpose of the other metrics? For instance, your method sometimes performs worse in terms of MBE (which, by the way, I cannot find a definition for in the paper). There are also some notable differences between the synthetic dataset and the real-world dataset for the axiomatic metrics; I encourage you to comment on why these differences arise. And why are some values for CAAE missing in Table 1? And since you are anyways considering so many metrics, why do you omit mean squared error which seems like a natural choice?\n- I would appreciate more elaboration on the setting, particularly what makes your method unique to time series. Could I not simply think of $H$ and $Y$ as multivariate random variables and ignore the time structure, or is there more to it?\n\n## Adjustment which I think could improve the work\n- Regarding your claims about minimising the upper bound of the entropy, I would appreciate a discussion of the conditions under which this approach is effective, and when it might fail.\n- I understand that the real-world dataset is proprietary, but it would be helpful to provide at least some information about it (such as sample size, dimensionality, etc.) to give readers a better sense of how it relates to the other datasets. Ultimately, what question are you trying to answer by applying your method to this dataset if you do not have access to ground truth?\n- I understand that the real-world dataset is proprietary, but it would be helpful to have a bit more information about it such as sample size, dimensionality and so on to give a bit more of an idea how it relates to the other datasets. Ultimately what is the question you're trying to answer by applying your method to this dataset if you don't have a ground-truth?\n- I would appreciate some more explanation of the causal graphs in Figure 2: what do the edges between H and Y represent since these are time series? Do all nodes inside $H$ have arrows going into all nodes in $Y$? Can nodes in $H$ also point to other nodes in $H$?\n- It would be helpful if you can further elaborate why only autoencoders are considered in this work? You state these are the “most extended and effective,” but it is unclear what this means.", "broader_impact_concerns": "No remarks", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "x8zLFsRKXN", "reviewer": "Reviewer_ah7J", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This manuscript presents a new framework for counterfactual inference in time series data, introducing the Conditional Entropy-Penalized Autoencoder (CEPAE). The study tests CEPAE on synthetic, semi-synthetic, and real-world datasets, comparing it against established baselines including LSTM forecasts, CVAE, and CAAE.\n\nThis paper makes a meaningful contribution to the intersection of causal inference, time series modeling, and representation learning.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The CEPAE framework offers an alternative to variational and adversarial architectures. It demonstrated empirical improvements across several metrics. The theoretical justification for entropy penalization as a route to causal disentanglement is particularly interesting. I think the paper offers sufficient evidence in showing the CEPAE framework.", "audience_yn": "Yes", "audience_explanation": "The paper advances the field of time series counterfactual inference with a novel and relevant methodological contribution. I think TMLR's audience would be interested.", "requested_changes": "While the authors acknowledge the challenges of comparing against synthetic control or causal impact methods, this omission weakens the empirical claim of superiority. Even limited comparisons (e.g., using standard Bayesian Structural Time Series) could provide stronger grounding for CEPAE’s advantages in practice.\n\nThe manuscript acknowledges the impossibility of full identifiability in multidimensional counterfactuals (as per Nasr-Esfahany & Kiciman, 2023: https://arxiv.org/abs/2301.09031). Nonetheless, it would benefit from a more explicit discussion on the interpretability of how the learned latent variables relate to domain factors and whether CEPAE’s disentanglement produces semantically meaningful components.\n\nWhile the paper notes that CEPAE avoids adversarial instability, there is little quantitative discussion of its computational cost, convergence behavior, or hyperparameter sensitivity. These are important factors for reproducibility and industrial scalability.\n\nThe real-world evaluation lacks ground-truth counterfactuals, making interpretive confidence somewhat limited. Although the authors employ proxy metrics, a qualitative example or case study illustrating CEPAE’s behavior in an industrial dataset would substantively strengthen the paper’s empirical narrative.\n\nThe exposition is dense and assumes substantial background in both causal inference and information theory. Several theoretical sections could be streamlined or visually summarized to improve readability. The core conceptual differences among CVAE, CAAE, and CEPAE could also be more distinctly contrasted.", "broader_impact_concerns": "The paper would benefit from:\n\nbroader empirical baselines beyond the autoencoder family,\n\nclearer articulation of interpretability and identifiability implications, and\n\na more accessible presentation of the underlying causal theory.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)