A Linear Algebraic Framework for Counterfactual Generation
Authors: Jong-Hoon Ahn
Akshay Vashist
Venue: ICLR 2024 poster (2024)
Keywords: counterfactual generation
individual treatment effect
synthetic data
Gaussian mixture model
Primary area: causal reasoning

Abstract: Estimating individual treatment effects in clinical data is essential for understanding how different patients uniquely respond to treatments and identifying the most effective interventions for specific patient subgroups, thereby enhancing the precision and personalization of healthcare. However, counterfactual data are not accessible, and the true calculation of causal effects cannot be performed at the individual level. This paper proposes a linear algebraic framework to generate counterfactual longitudinal data that exactly matches pre-treatment factual data. Because causation travels forward in time, not in reverse, counterfactual predictability is further strengthened by blocking causal effects from flowing back to the past, thus limiting counterfactual dependence on the future. Using simulated LDL cholesterol datasets, we show that our method significantly outperforms the most cited methods of counterfactual generation. We also provide a formula that can estimate the time-varying variance of individual treatment effects, interpreted as a confidence level in the generated counterfactuals compared to true values.

Published as a conference paper at ICLR 2024
A LINEAR ALGEBRAIC FRAMEWORK FOR COUNTERFACTUAL GENERATION
Jong-Hoon Ahn & Akshay Vashist
Otsuka Pharmaceutical Development & Commercialization, Inc. Princeton, NJ 08540, USA (jong-hoon.al) ahhn, akshay. akshay.vashis
vashist st @ot.sukaauuue casssuukkuus.cc com
ABSTRACT
Estimating individual treatment effects in clinical data is essential for understanding how different patients uniquely respond to treatments and identifying the most fective interventions for specific patient subgroups, thereby enhancing the precision and personalization of healthcare. However, counterfactual data are not accessible, and the true calculation of causal effects cannot be performed at the individual level. This paper proposes a linear algebraic framework to generate counterfac- tual longitudinal data that exactly matches pre-treatment factual data. Because causation travels forward in time, not in reverse, counterfactual predictability is further strengthened by blocking causal effects from flowing back to the past, thus limiting counterfactual dependence on the future. Using simulated LDL cholesterol datasets, we show that our method significantly outperforms the most cited meth- ods of counterfactual generation. We also provide a formula that can estimate the time- varying variance of individual treatment effects, interpreted as a confidence level in the generated counterfactuals compared to true values.
INTRODUCTION
Estimating individual treatment effects (ITE) within clinical data is pivotal in advancing healthcare towards more personalized and effective treatment strategies (Simon & Perlis [2010: Kent al.) 2018, Hoogland et al., 2021 Berchialla et al., 2022). It allows a nuanced understanding of how diverse patient populations uniquely respond to various treatments. This detailed insight crucial for pinpointing the most beneficial interventions for specific patient subgroups, thereby significantly enhancing the precision and personalization of medical care. By tailoring treatments to patients' individual characteristies and conditions, healthcare providers can improve treatment outcomes, minimize adverse effects, and ultimately elevate patient satisfaction and quality of life.
However, a significant challenge in this endeavor is the inaccessibility of counterfactual intu-innfrmationn on what would have happened to a patient had a different treatment been adminis- tered. Without this data, accurately calculating the true causal effects of treatments at the individual level remains an infeasible task (Holland) 1986). This limitation hinders our ability to make definitive statements about the efficacy of one treatment over another for a specific patient. The absence of counterfactual scenarios in clinical data necessitates the development of sophisticated statistical models and methodologies (Shalit et al.) 2017; Lu al. 2018 Yoon et al. 2018) that can estimate these effects as closely as possible, using the available factual data and making informed assumptions about the unseen counterfactuals.
This paper proposes generative model that can predict counterfactual data given observed longi- tudinal data, including multiple pre-treatment measurements. The use of multiple pre-treatment measurements in randomized clinical trials has been proposed in recent years (Van Patten et al., 2002) Vickers 2003 Ma & Wang 2023). Pre-treatment data are measured multiple times before participants enter clinical trial for several key reasons (Friedman et al. 2015). Firstly, collecting data at various points can help establish more accurate baseline of the participant's 's health status. By averaging these measurements, researchers metrics, providing a clearer picture of the participant's 's condition before the intervention. Secondly, multiple pre-treatment measurements allow for the assessment of the stability or progression of
mitigate the effects of anomalies fluctuations in individual health

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condition over time, offering valuable insights into the natural history of the disease being studied. This information is crucial for interpreting the effectiveness of the treatment, as it helps to distinguish between changes caused by the treatment and those that might have occurred naturally. Lastly, but more importantly for our approach, these repeated measurements before intervention can improve the precision of the study's outcomes. By using these baseline values covariates in analytical models, such as ANCOVA (Frison & Pocock, 1992), researchers can control for individual differences at the outset, thereby increasing the study's 's statistical power. This approach helps make more accurate inferences about the treatment's 's effect, ensuring that the observed changes are indeed attributable to the treatment rather than to pre-existing trends or variability in participant's 's conditions.
There have been efforts to use multiple pre-treatment measurements and their inherent temporal
structure for counterfactual predictions (Abadie & Gardeazabal] 2003; Xu 2017] Amjad et al. 2018, Qian et al.) 2021] Doudchenko et al.) [021; [Shao et al. 2022). However, most do not find counterfactual data whose pre-treatment data exactly matches the pre-treatment factual data. This paper introduces linear algebraic framework to create counterfactual longitudinal data that aligns perfectly with observed pre-treatment factual data. The essence of this approach lies in its acknowledgment of the fundamental principle that causation operates in forward temporal direction, not backward. This framework significantly enhances the predictability and reliability of counterfactual outcomes by preventing causal effects from retroactively altering past data, thus limiting the influence of future events on counterfactual scenarios. This methodological advance helps accurately model outcomes under various treatment scenarios, enhancing causal relationship insights. Such a framework is valuable not only in understanding clinical data but also in any field where understanding the precise impact of interventions over time is critical, enabling a more informed approach to decision-making based on the potential future effects of actions taken in the present.
2
METHODS
2.1 LEARNING STATIC-STATE REPRESENTATION FROM OBSERVATIONAL DATA
Our framework of counterfactual generation starts with simple assumptions.
Assumption (State), A real-world unit subject to an action, treatment, or regime is represented by static state M
Examples real-world unit may include a physical object, firm, a patient, an individual person, or collection of objects persons, such as a classroom an alternative action, treatment, or regime has a different static state. For our purposes, the same physical object or person at different time remains the same unit Instead, unit is observed in time-varying form by the next assumption.
market. The same unit exposed to
Assumption 2 (Observation). A real-world unit represented by static state s can be observed by time-varying observer W(O,
x
w
where 1 (t) is a measurement noise.
This state-space representation model may fall under the specific category variant systems) of the general state-space representation of linear system in the control theory (Brogan 1991).
2 (explicit discrete time-
The measurements made by linear observers are referred as observational data.
Definition 2.1 (Observational Data). (t) e RD is the observational data of s at time t. D is sum of the number of outcomes of interest and the number time-independent or -dependent covariates.
Suppose that we have N units' longitudinal data, D = {x E I E (0,1); = 1. 1.... T:
1,..., ,N), where Xn,w,, is observed data of the n-th unit who received a treat- ment Wn E 13. The number of units that received w, = 0 is No and the number of units that
It was assumed that the same object or person at a different time is a different unit by Imbens & Rubin 2015); Yao et al.) (221))
"Specifically, the system matrix A (*), the input matrix (1), and the feedthrough matrix ( are zero while
the output matrix (1)) is w
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received wn is N1. which sum to N No + N1. Wn = 0 or wn 1 can mean that the n-th unit belongs to a control or treatment group, respectively. Generally, each treatment may involve fixed combination of multiple medications. For example, Wn, 0 wn with three different medications of {A, B, C} and 7 = 5 can mean that the n-th patient takes A ABCB BA or A ABAT BC, respectively, where the first different medication is found & = 3 and then we say the intervention time I is To 3. If the n-th patient took treatment Wa = 0 meaning ABCBA and we obtained its factual outcomes, what would be probable outcomes if the patient had taken treatment wn = 1 meaning ABABCE We can regard the first To outcomes as the same to the observational data, but it is challenging to predict the next outcomes. The binary treatment cases are an example of the counterfactual generation problems that we try to solve, and it is straightforward to extend it to three or more treatments cases.
Assumption 2 raises representational learning problem of how to find static states from those N units' observational data by solving system of T equations given by Eq. (T). this paper, we propose to use ca-conditional Gaussian mixture model (GMM) as a prior probability distribution over the static states by
K. Σ
p(s(w)
Nea Nig (s) Mkww Σx,w)
where Th Mk and Σt are the mixing proportion, mean vector, and covariance matrix of the k-th mixture component, respectively, for w {0, 13. Eq. implies a probability distribution over xf0) -saace for given s of the form
pxx(9 (9 8) = (2m)= 3 v. - exp { 1 1x(00 7 (x!0) w(Oss)
(3)
where V. E IR DxD is a diagonal noise covariance matrix such that n NO0, V.). We can estimate those parameters of Q = {TTR, Mi.. Iy.., W(O, v E f0, 1 = 1,... T 1, 1.... 3
and static states s by the EM algorithms for maximization of the complete data log-likelihood
N T LC = -E 1=1=1
where the rithh unit's state Sn.,,, is regarded as missing data and given as an expectation (Sm by the EM algorithms derived in Appendix A. We refer to solving Eq. T by using the EM algorithms as a static state analysis with priors of w-conditional Gaussian mixture model or SSA-GMM.
Equation is replaceable by any other model long as it universally approximate the underlying data distributions, but in this paper we use a GMM as a universal approximator of the state densities.
Assumption 3 (Universal Approximator). A Gaussian mixture model is universal approximator of densities, the sense that any smooth density can be approximated with any specific nonzero amount of error by Gaussian mixture model with enough components (Goodfellow al. 2016).
2.2 GENERATING COUNTERFACTUAL DATA
Counterfactual thinking is imagining how things could have been different if something had changed in the past or future. If different action, treatment, regime had changed the state s of unit by Assumption| I the observational data would have been different through the observation of state s by Assumption 2 To get the counterfactual data, we need to imagine that the unit had been observed under the same-world conditions from which the observational data were obtained.
Assumption 4 (Counterfactual Thinking in State-Space Model). When a unit received treatment w, we obtained its observational data x by observing its state sw at time f, L.e.
((0) s.. + n (t)
If the unit had received an alternative treatment w. we would have obtained its observational data x through the same observational process for its state sg, i.e.,
w S + n (1)
which is counterfactual conditional statement for the unit.
'The same notation, To, for intervention time was used in Abadie et al. 22000) Abadie 2021
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Since effects do not precede their causes, any counterfactual data observed from different treatments at time To cannot differ from the factual data prior to To. We should ensure the independence of an intervention made at To by using the first To observers. Given a sequence of the first To observers {W(*) E IR Dx M ID < M, 1 S S To}, we let N(To) E RMxx be an orthonormal basis for the null space such that ( = O for S To- To ensure no backward causation (Lewis, 1979), i.e., X x
(t) for To, we need ( (8s - s.) 0, which means that the difference two states must lie in the null space spanned by N(To).
Definition 2.2 (Counterfactual State). Suppose that we generate So p(s(@) from Equation subject to
mw = TT)
(5)
with an arbitrary vector & RR Then, we call 8. a counterfactual state that the unit would have if had received the alternative treatment W.
By the nature of a counterfactual state defined by Definition 2.2 counterfactual state reasoning is a random process defined as a sequence independent and @-conditionally distributed random states
So of Definition 2.2 Generating counterfactual data under each treatment @ are deterministic process of the observation of each random state, as described in Assumption To understand Eq.
(5) let us recall:
called an affine subspace of R all affine Definition 2.3 (Affine Subspace). A subset U C IR
combinations of vectors in U remain also U:
J+1
J-1 Σ
u1, UJ+1 EU, A1, ... A141 E with
Σ j=1
23 =1 =
Auu E..
j=1
If we put u, [N(To)], + sw for j = 1.... and u,41 = Sw with X, = En for j 1, and
AJ+1 - -₁= Eg, we can find that s, E U from Eq. 5
The use a GMM in Eq. enables us to directly generate counterfactual state. For that, we need two lemmas:
Lemma 2.1 (Conditional Distribution of a Multivariate Normal Distribution). Given a multivariate normal distribution N (s|pu, Σ) and an affine subspace Un = (s s So WoE, & } with Wo MJ and M > J, the conditional probability density function p Els Uo) is also normal distribution and is given by NeEmm, C) where the conditional mean and variance are
m = (W W.) C = -WWTT (wo)
so)
Lemma 2.2 (Conditional Distribution of Mixture of Multivariate Normal Distributions). Given mixture of maltivariate normal distributions En TrN(s) Ix) with E Th and affine WoE, & E R.) with Wo E BRxx and M > J, the conditional
subspace Uo {s lss-- probubility density function P(E E Up) is also a mixture of normal distributions and given by Ex P:N((NEm, Ck) where the conditional proportions
TWN(Wom; + so soluk C
Σn TWWW (Wom. + Sol Ix C
with
my = (w/ ( - (M4 So)
C =
-( (W. -
Finally, we make our main contribution to counterfactual generation:
Theorem 2.3 (Gasussian Mixture Counterfactual Generator). Let be maximizer of Equation (4) for a factual dataset Adding (()) En, to Xn.a, generates counterfactual data of the n-th unit.
(0) (()
Xawa W(NN(( En.on
(6)
4

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space N
Observation space
Sreatment
factual
EM algorithm
factual state (Smm
Sampling along spacs
2. Sampling along space
Scontrol
counterfactuaf
3. Projection
(t)
Sactual (Sooae)
W(
(Tg
(Tn) X Ru,0
Nyp SRAA
W
Figure 1: conceptual view of how synthetic counterfactual longitudinal data are generated.
where En,on a random vector such thut
Kon pk N(E/mR,,n Ck.
Ln
(7)
with
Ck.an
(N(TT)TI- I-1 N(To))-1 K
(8)
Cka. (To)T
k.an (Hk,o.. (Sm,e)))
(9)
Tha ((N( (T)) mg,a. (Spnee) Mk. Ik,o,) (Ck,,
Pk,aa
Σken The THA (NN (T))mmr miro, + (8n,us aa. IMc,, Cke a,
(10)
Corollary 2.3.1 (No Backward Causation). Any counterfactual data x do not deviate from the factual data Xn. until the alteration of treatments at / = To-
Figure [ [ssows a conceptual summary view of counterfactual generation. Curved data manifolds denoted by Scontrol and Smeat are estimated as p(s(w = 0) and p(s|w 1) of Eq. @ respectively, by maximizing the log- ligeliiel function of Eq. in terms of GMM parameters proportions
Tkws, means M and covariances Σk... We can generate synthetic factual data using their own GMM parameters, which is not of primary interest this paper. For a counterfactual generation, from the (blue) factual point on Streat, for example, we need to sample out new (red) point Sn.w, from a GMM of the opposite manifold Seenuot- which is annotated with "sampling along null space" and performed by using Eq. (5). Eq. (5) includes random variable & sampled from the conditional distribution, Eq. 7 After randomly generating a counterfactual state Sn,D, we can finally obtain the time-vane varying counterfactual data projecting the fixed point Sn,,, onto the moving lower-dimensional observational space spanned by (!)) or, equivalently, using Eq. [.. Here, any generated counterfactual data are equal to the observed factual data at < Tu, which is stated in Corollary 2.3.1 After t To. may deviate from the factual data.
2.3 ESTIMATING INDIVIDUALIZED TREATMENT EFFECT
In order to estimate an ITE, we need read out the outcomes-only difference because x E R includes covariates as well as outcomes of interest for the n-th unit at time t. The individualized treatment effect (ITE) for the n-th subject after the treatment time To with w,, [0,1] 1] is defined as:
ITE(1) (t) (1 2wnn 2ww)w" W (x
effect (x,,0, x!) (
(11)
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(1) where x is synthetic counterfactual data of the n-th subject's factual data, Xi,ws Weffect is column vector to return an ITE selecting one of the differences between factual and counterfactual data. For example, it could be [1, 01...... ,0'' if the first component of x is the outcome of interest. Corollary 2.3.2 (Individualized Treatment Effect), Let Q be maximizer of Equation 4 for factual dataset D. The individualized treatment effect through an effect operator Weffeet is given by
ITE = (1 ) W ( (To)) En
(12)
where En,a, is given by Eq.
77
Remark 1. Equation 12 can be read as
(TE(1) Weffeet W (1 N (readout) (observer) o (treatment effect).
Equation gives a random value. We can also calculate its mean and variance.
Theorem 2.4 (Expectation- Variance of ITE). Let Q be maximizer of Equation (4) for factual dataset D. the expected value and variance of individualized treatment effects for the nith unit are given by
Kwn Σ
[I
(1 2w0n) offfect
((b)
Pk,a, Mkk.
(13)
Kan
Var ITE
Csottl k,oo
N (((l Weffect
(14)
where
dotaa kian
wiithin kiW
hetween Chinn
Ck..a, + (mam... - ma,))mga -ma.)
Σ
M
Pk m
3 EXPERIMENTS
3.1 CALIFORNIA CIGARETTE SALES DATA
We demonstrate Corollary 2.3. and Thcorem by using a public example: the problem of estimating the effect of cigarette taxation per-capita cigarette consumption. In 1988, California introduced the first modern-time large-scale anti-tobacco legislation in the United States. To analyze the effect of California's legislation after 1988, use the annual per-capita cigarette consumption the state level for 38 states not adopting anti-tobacco legislation programs between 1970 and 2000.
The data set has T 31 (years of observation), N No + N₁ 38 (untreated states) (California), and D 1. For analysis, we set (M, Ko) to be (30, 2) and trained our SSA-GMM method for only control group data of 38 untreated states. The expectation curve the synthetic counterfactual data of California is estimated by using Eq. (13) and is plotted in Fig. 2 We can observe that the pretreatment synthetic California data exactly matches the factual California data between 1970 and 1988, as described in Corollary 2.3. Our method is random generation, and the standard deviations are plotted with pink filling, which is computed using Eq, (14)
The mean counterfactual curve comes from the convex combination of Ko centers, denoted by m with nonnegative and sum-to-one weights, denoted by Pak: as described in Eq. (13). Interestingly, using convex combination of nonnegative and sum-to-one weights is also the original key idea of the synthetic control method (Abadic & Gardeazabal 2003).. Accurate pretreatment reconstruction by the synthetic control method (SC) any SC-based method usually gives overfitted post- treattrett result. contrast, our post- treattetetment result shows no overfitting despite the pretreatment fitting and a convex combination.
There are slight differences between our post- post-treatment prediction and the results the synthetic control method after 1994. Which is more cannot be determined; it agrees with the Bayesian

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per-capita cigarette sales (in packs)
120
100
80
counterfactual Califernia (SCA-GMIM)
counterfactual California (SCM)
factual Califoriaa
1970
1975
1980
1985 year
1990
1995
2000
Figure 2: Application of SSA-GMM to California cigarette consumption data.
approach result of Amjad et al. (2018) showing that it may be possible that the synthetic control method overestimated the effect of California's legislation. Evaluating how much true synthetic coun- terfactuals are in ITE estimation requires simulated dataset, including ground truth counterfactual data, in the following section.
3.2 SIMULATED LDL CHOLLESEROLL DATA
used the simulated To evaluate our counterfactual generation method provided by Theorem 2.3
LDL cholesterol dataset that had been used to evaluate the SyncTwwin algorithm (Qian et al. (0021). effect of statins, a drug commonly prescribed to
We aimed to estimate the LDL wwolesterol- hypercholesterolaemii patients. The dataset is created using the widely adopted pharmacology model in the literature (Faltaos al., 20066 Kim al., 201T). For more information about the simulation study, refer to the paper of Qian et al. (2021).
Table 1: Mean absolute errors (MAE) on ITE under different levels of confounding bias po and data
size No. is assumed that every unit received treatment at To 25 and there are no missing data. Estimated standard deviations shown in parentheses. The best performer is in bold.
N No + Ni
400
N No + 1000
200 1200
Method
Po = 0.1
Po 0.25
Ph = 0.5
Pn 0.1
= 0.25
Po = 0.5
SSA-GMM
0.072(.004)
0.072(.004)
0.076(.004)
0.070(.004)
0.073(.004)
0.069(.004)
SyncTwin
0.308 (.037)
(.003)
0.116 (.008)
0.178 (.012)
0.106 (.006)
0.094 (.005)
SC
0.341 (.042)
(.024)
0.150 (.018)
0.231 (.033)
0.172 (.03)
0.158 (.023)
RSC
0.846 (.046)
0.357 (.021)
0.299 (.018)
0.296 (.016)
(.084)
0.292 (.015)
MC-NNM
1.160 (.060)
0.613
0.226 (.012)
0.526 (.029)
0.177
0.124 (.006)
CFRNet
0.888 (.076)
0.166
0.279 (.027)
0.236 (.018)
0.139 (.008)
0.113 (.007)
CRN
0.530 (.035)
0.631 (.048)
0.343 (.023)
0.456 (.027)
0.404 (.028)
0.371 (.030)
RMSN
0.646 (044)
0.608 (.040) 0.538 (.038)
0.720 (.053)
0.635 (.046)
0.601 (044)
CGP
0.674 (.043)
0.619 (.039) 0.552 (.035)
0.825 055)
0.690 (.046) 0.602 ( (.038)
To explain how and why our algorithm is robust to bias, we analyzed an outlier sample, which is shown the black dashed line in Fig. 3] the test dataset of 1. The dashed curve is represented by the yellow point of (s..). There are no similar curves in the w = 0 group test data, and the In our notation, the pre-treatment time steps go from post-treatment effects are estimated at 26 through f = 30. The dataset includes two covariates and one outcome, which yields = 3. The big difference between our approach and the SyncTwin method is that we also use the post-treatment data of covariates and pre-treatment data of outcomes. In contrast, SyncTwin Twin uses only pre- rreetreatment data of covariates and post- treattrettmntt data of outcomes. To produce Table as reproduction of Table 2 of the SyncTwinn paper (Qian et al. (221), we trained our model with Ko = K, and M = 85 from a dataset of No N₁ 200 and a dataset of No 1000 and N; 200. We also introduced confounding bias denoted by p.. for the n-th patient:
through
25, where To 25, and the
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N
CR E
N
(T
Eg,bhn
Mk,00
E-B
(snaar)
kg,0
Ak.
outcome: (cholesterol level)
outcome (cholesterol tevel)
outcome:
countertactual (synthetic)
counterfactual (oynthetic)
countertactual (nynthetic)
outcome (cholesteral levell
outcome loval)
nutcome Icholesteral levall
factual iground
Fectual iground trun)
factual (ground truit)
counterfactual
counter
time (1)
(cholesteral tevel)
time
outcome (cholestorol
fectual (groand inuto)
ouccome
fechusl (groand truit]
factual (groand trut)) (ground
(groudd
time (1)
time
tmm (?)
Figure 3: Outlier sample analysis. (upper) A conceptual view the k-th Gaussian component and sampling points A-/; (center) Test outcome datasets of group and group: (lower) Outcome curves generated by SSA-GMM from the points.
Pr. for wn 0 and Pr. = 1 for wn 1. The constant Po controls the degree of confounding bias (smaller Po. larger bias).
Table I shows our method's improved performance compared to the methods of SC (Abadie et al.
2010), RSC (Amjad et al. (2018), MC-NNM (Athey et al. 2021), CFRNet (as SyncTwin-combine form Shalit et al. (2017)) CRN (Bica et al., 2202)). RMSN (Lim 2018), CGP (Schulam & Saria 2017), and SyncTTwin Our model is especially robust to confounding bias Po- As exemplified in supplementary Fig. counterfactual pretreatment curves generated randomly by Eq. (6) and Eq. (7) exactly match their pretreatment factual data. Moreover, those generated counterfactual curves are susceptible post-treatment overfitting. A family of synthetic control methods directly exploring
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convex combination of real controls or transformed controls (e.g.. Equation (4) of the Tynn paper) does not achieve this property.
To explain how and why our algorithm is robust to bias, analyzed outlier sample, which shown as the black dashed line in Fig. 3 in the test dataset of w 1. The dashed curve is represented by the yellow point of (s.)) There
similar curves in the 0 group test data, our model, is given by (V(ss)) ( E/ In for any &, which is extremely thin. Most algorithms listed in Table fail to generate counterfactual to this outlier. However, the cut and normalized (or, equivalently, conditional) probability distribution N (E, Ct.o.), as given by Lemma 2.2 and cut by the null space (pink arrow), can generate good counterfactual (see the subplot F... Generally, cutting probability density function by the null space and normalizing the cross-sectional density function make it robust bias. For comparison, we generated curves on the in-sample points of A-F over the null space and the out-sample points of G-I.
and the probability density along the null space is extremely thin.
Table 2:
Mean absolute errors by SSA-GMM on ITE under different noise levels a. The second row the same as the first row of Table
N = No + N₁ = 200 + 200 = 400
N No + N₁ 1000 + 200 1200
Method
= 0.1
po 0.25
= 0.5
Po 0.1
= 0.25
Po 0.5
0.01
0.029 (.001)
0.019 (.01)
0.097 (.002)
0.026 (.001)
0.016 (.001)
0.014 (000))
0.1
0.072 (.004)
0.072 (.004)
0.076 (.004)
0.070 (.00))
0.073 (.004)
0.069 (.004)
0.2
0.183 (.011)
0.144 (.008) 0.146 (.008)
0.177 (.010)
0.143 (.008)
0.139 (.008)
The dataset used for producing Table to a dataset made by solving the PK/PD model. Table shows results by different noise effects with a 0.01 and a 0.2. Our algorithm's MAE (0.183 and 0.144) 0.2 are better than Synn'Twin's 's MAE (0.308 and 0. 150) at = 0.1, which strongly suggests that our method is robust to additional noise.
created by adding additional noise \O0 0.1)
CONCLUSION
We have introduced the static state analysis with Gaussian mixture model (SSA-GMM) and its use as generative model for synthetic counterfactual longitudinal data. We have also demonstrated its ability to generate a counterfactual of -ndividual-level observed data. Importantly, can estimate the individual treatment effects (ITE) that change only after treatment using the synthetic counterfactual outcomes data or the formulas of expectation and variance ITE. Using LDL cholesterol datasets that we simulated from PK/PD model, we have shown that our method is robust to confounding bias and noise, greatly outperforming the most cited methods of counterfactual generation and ITE estimation.

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ACKNOWLEDGEMENTS
We would like to thank Hossain Saboonchi and Anton Palma for their useful discussions and feedback on the early draft of this work, and Ksenia Korshunova for helping experimental work.
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A DERIVATION OF EXPECTATTION-MXXIIIIZAIII ALGORITHMS
A.1 STATIC STATE ANALYSIS WITH A GAUSSIAN MIXTURE MODEL
For the case having noise n (O, P. with a diagonal matrix w. E DDDD Eq. implies probability distribution over (1) for given IR of the form
psx (2x) 2 |
exp {
2 1x00
-
ww. -W00))
(15)
With a Gaussian mixture prior over defined by
p(s) M Ix))
(16)
we obtain the marginal distribution of (9)) in the form
exp 2 (x(0 - To (x(0) -Woo (17)
1)x
Σ
Th 2T C2
k=1
where the model covariance is C/ = v. + I W(OT.
However, the joint distribution of all the observations fx(1) ... x(7)) is intractable with
p(x
x!))
((x x(T- x (T) )p(
p(x(1), ......., x (T x (T x )x) )((x)))
Let us define x and W by
... = 1(x0)) (x..))T... (0 (x())T E RTD
(18)
= (wwi)y (w(+)T. ... (w())).....(")))) (
(19)
Then, obtain the conditional distribution and the marginal distribution in the form of
1 -x - Ws)}
p(x(s) 12 exp
(20)
p(x) Σp(x(k)p(k)
Σ Th [2 exp 1-10-- WHa) (x WWi)..
1
(21)
where C = V + WE, and v - & v. E (TD))x(TD) by the Kronecker product &. By Bayes' rule, this leads to the posterior distribution of the form
p(s(x)
E Σ pkk]x) p(xls)p(slk)//pxxI)
- 2TMa exp { - ((s ((s- m4 M.W. (x WHel)"
M ((s 14) M.TTy WHa))
(22)
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where M2 (Ix²² WWy
Now suppose that we have observational data xx1) for n = 1, ,N and I...., We can
maximize the log-likelihood function of Eq. 44 for the joint distribution of observational data x!! and the latent variables S, in the form
44 (xn Wen) x
sn)
|2 exp
- (xn
Th
2T x1 exp
{
-
1
(sn - -1/) 2x (sn - me) }
(23)
2
A=1
By calculating the expectation values
Ynk p(k(xn)
(24)
(8n)k / p(s) k)) lds
(25)
(sn) / p(s)x,) lds)
(26)
((sn - (sn))(sn - (8m))
Σ xk))))) Mc.
(27)
we can obtain the expectation of the complete data log-likelihood the form of
K
1
1 T
(Co)
Ink
(In Th
In |Ix|
ur( (s,
+ p I (s.)
2 uu
I
H
2
n=
1=1
In 4
x
Typ
xn
+ x
W (sm.)
tr
4T -W WW(ns??))
2
2
2
K
A
Tk
(28)
where the expectation values are given by E-steps:
(x WHI. Ct)
Yin
Ex Tx,,N, Ct:
(29)
(sn)
H4 + M. M. 'xx, WHz)
(30)
(sm) Σ Yakk (sn)
(31)
(s,, Σ Tnn (M₃ + ((8n)x - (sn)))(sn)> (sn)) + (8m) (s..)
(32)
with C₁ W WE, WT and M = -I + www 'w)
Equation 28 is maximized by the following M-step formulas
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M-steps:
En Yink
7
(33)
N
Hk
En Yink
Σ Yrkk (sn)
Emik
(34)
Is
Σ
Yon (88
M4 (8m)
(8n) + H
(35)
En Yink
w(1)
Σ (sx)
Σ (S
(36)
v.
Σ
trp
XxxX
(W.)s + (s
(37)
NT
where trp ! is a d x d diagonal matrix defined in this paper as returning diagonal matrix
0
0
0
0
trp V
(38)
0
0
0
0
(o.DDD DD/
with bhd IT-1 v (DL+d)(Detd) Notice that tr 4 is equal to the trace of V
A.2 STATIC STATE ANALYSIS WITH A CONDITIONAL GAUSSIAN MIXTURE MODEL
Now, we are ready to derive the w-conditional EM algorithms, Eq. 28 should be rewritten as
K N
(Cc) =
Iak (In Thww
In [ [kwunl
2 tr I kwn
(Sm,w, ST )) + M4 In kyaan (Smww))
2
Hkwa ke Heawa
4
w-lxx Xn.wn + xT,w 4 (Smmun)
2
2
Kun
1 tr(w"y wr WW (Sm,u, ST s..m 1)) A
Tha,
(39)
2
where the E-steps are given as:
E-steps:
Thk (Xn.wa W WHe CR.wa)
Yink
(40)
Σ Ln Th ann N (Xenenn [W WHag aam : Ck' Cor
(sn dik
Mkwwn
M Me.eo 1 WHe
(41)
Kun
(sm,wnn
(42)
k=1
Kwn
(Sn (Tim))
Σ
Ink (Miwnn + ((smaon do (8m (S5 (Sm ))²)
+ +Sn.a. (Snaon))
(43)
with Ck. = + WE WT and ME (Irl + WTY -yTy-WW)- for = 1,... K with w E {0,
Equation 339 can be maximized by
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M-steps:
En Swann Yokk
(44)
En Ouswn
Σn Owan Yink (Sm
Hkw
(45)
En Buss Yu'k
Σn Ownn Yank (Sm stiwn Mkw w (8m (Sm.a 14 + uk Hk
(46)
End Ownns Yn'k
W
Σ Xniun (Smmaa) yJ
Σ (n spann
(47)
V.
NT trp Xnjon X Xnwn.
2W (Sm X + W (Sm,s, If n,w
W
(48)
where Swnn is 1 for w = wn or for w + Wn-
A.3 STATIC STATE ANALYSIS WITH INCOMPLETE DATA
Suppose that x defined by Eq. 18 is incomplete data and we separate it into
pobs
miss mis
(49)
where ths and mis are observed data and missing data, respectively, with dim(x"">) + dim(x""i)
= TD. We assume that robs and pmis are constant matrices only having elements or such that
obs
robs T x
(50)
pmss T X
mis
(51)
or, equivalently, ross pobs I, mis
mis
I, and pobs pmss
O.
The noise covariance matrix V is diagonal, and the conditional distribution is separated as p(x(s)
p(xo s) (x) in the form of
p(x
y
exp
(x bbs
s) Tyros -1 "x Wuubys)
(52)
p(xmisls) (s)
12 ymmis
exp
mis
Wiiss)) Tyrmis -1 (x Wimss
(53)
where yrobs
pobs? Trr ymmis
pmis T y Wros robs? T W, and Wmis pmis W. With
a Gaussian mixture prior over the latent variables defined by Eq. (16), we obtaín the marginal distribution of observed data in the form of
0bx
K
12 C exp {
1 1 (x -Wobs C
(x W wrobs
(.))
(54)
where Clobs ypu the form
Wobi E kuw. Wobj""
By Bayes' rule, this leads the posterior distribution of
p(s|x oha
Σ
12
exp (s (8)))) M (s (s)s)
(55)
k
where
Mohs
(I,") + W yots 'Wobs-- -11
(56)
(s)
H4 Mobo wrobs yoots (x Wobs H4))
(57)
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Equation 28
is rewritten
Kwn
1
1
(Cc)
In Tkin
In | k,ool
tr(=
tr(2
kieen
(m ww 1) Mk Iksen (Snauee)
2
kkwa I k, Mkian
1 oheT you
nbs Xnaan
oheT yobss X
Waits (sm,am)
2
n,wn
tr yyohs "Wobs (Sn
In 14 Yink
Thww -1) (58)
where xna
robs
Wobs
robs
W. youhs
robs T TTo and V ② V.
Equation
(58)
is calculated by using
E-steps:
a M(xobs
W M,n K,an,
Pink
Σx ky TK, (xxobs
Wobs Me
C
(59)
(Sn,wrh
Mk
Mohs
Waht ypoks
Wobs Mk
(60)
Kwn
(Sm,wn)
Ynk (n
(61)
k=1
Kun
(Sn,wnn ST.wen)
Iun (M + (Smmaon.k (8m (S (8m ())
k=1
+ (sn,wa (sncom)
(62)
with C,u,,
ynni
W I Inas,, Wob and Mobs
(2 k,ws
Wobs ypnhs Wobs
-1
kewin
Equation 558) is maximized by using M-steps:
Swwn Ynk
Tk
En Senwrr
(63)
Σn Swen Yook ( (sn,on)
(64)
Eny Oww Yr'k
Σn Ownn Yrkk (Sn,wr ST. )
M4 (sm,on)) (Sn uu + ucH4
Ike
(65)
En Swnny Trnk
Wr
Σ
An, nobj ooyj ny neain (sm,un,) yz
Σ Anr S stiwen)
(66)
v.
tTD
robi
bby
trD
robj moby?"
(67)
where w, and robs are the r-th rows of W and robs n,r
respectively, and
A, pobs pobs?
diag(An.1.) An.rs...)
(68)
A = x xna
objT wobs (Sn x xbj xna
(Wobs (sn....)) ))
Wobs (Sn,s, smann)
Wobss"
(69)
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The posterior distribution of mis is given by
/ [ s/ s|xen) = f ds (ussp(xee / s)p(s((00t)
p(x (xb)))
/ [ss|p (s) = / Idspp(x ss) Σ p(k: xe (p( (p(s) k)
K
Σ p(kllxtm) 12T2k
exp 2 (xxtix
mix
(x (x )k)² Z
1=1
(x (xx (x) }
(70)
where
M
red
(Wriis
ymis
Wimss
+ Mu )
(71)
Wmis M Wmis Wimss ) -1
Z
ypmis
yyrnis
yomis
(72)
(x mix
Z miigpmiis
Wiiis
M M
M (s)
(73)
We can also estimate the missing data by
(x k
Z
Zass ymmis Riwn
Wiiis
M Jobs
M (s )
(74)
Kwn
(x
Yokk (xxmis )
(75)
Kwn
(x
mis
mist Xnnwwa
Yink (ZRis
+ ((x)) (xx (xxmin)))) )x - (x ))
1=1
+ +x
(xemis ST.
(76)
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B SUPPLEMENTARY RESULTS AND DISCUSSION
Supplementary Results.
Table I is reproduced one of Table 2 of the SyneTwin paper. Table I itself shows our state-of- art results across different biases and data sizes. The training/testing dataset used was added by independent white noise (0, 0.1). Our results of mean absolute enror are the level of 0.07, which means the ideal performance that cannot be algorithmically more improved when we notice
wwwp:yaabsnnp..anomm..mam (10000000))) 0. meann) = 0.07979.
Tables and 2 are also supporting that method is robust to data missingness and noises. As requested by the Reviewer, we performed more experiments (including the suggested MC-NMM and CGP) under diverse treatment times and over prolonged post-treatment durations follows:
We also tested algorithmms for incomplete data of 50% missingness, and Table 3 shows that our method is robust data missingness. SyncTwin's 's performance gradually decreases by the missingness rate and the training data size, whereas SSA-GMM keeps its performance (compare No 1000 results of Table [ and Table (3) and drops at around Ng 200. The EM algorithms for incomplete data derived in Appendix A.
Table Mean absolute errors by SSA-GMM. SyncTTwin and SC on ITE for incomplete data of 50% missingness. The best performer is in bold.
N No + N₁ 200
200 100
N No + N₁ 1000
200 1200
Method
Po 0.1
Po 0.25
Pu = 0.5
Pu 0.1
Pu 0.25
Pu 0.5
SSA-GMM
0.327 (.088)
0.232 (.016) 0.141(0009)
0.091(.006)
0.054(.004)
0. 0.061(.004)
SyncTwin
0.351 (.038)
0.180 (004))
0.148 (.011)
0.202 (.018)
0.142 (.010)
0.113 (.007)
SC
0.441 (.044)
0.272 (.024)
0.252 (.030)
0.424 (.041)
0.298 (.028)
0.228 (0.021)
Table 4: Mean absolute errors (MAF) on ITE under different lengths of the temporal covariates (or intervention times) To and data sizes No with missing data and pn 0.5 confounding bias level. Estimated standard deviations are shown in parentheses. The best performer is in bold.
N No + N₁ 200
200 400
N = No
Ni 1000 200 1200
Method
To 15
25
To 45
To 15
To 25
To 45
T =
30
T 50
20
T 30
T = 50
SSA-GMM
0.092(.005)
0.076(.004)
0.081(.004)
0.065(.003
8.069(.004)
0.081(.004)
SyncTwin
0.121 (.009
0.128 (.008)
0.120
0.097 (.005)
0.094 (.005)
0.085 (.004)
SC
0.140
0.149 (.018)
0.138 (.021)
0.190 (.029)
0.214(.006)
0.215 (.044)
RSC
0.348 (.023)
0.322 (.019)
0.228 (.011)
*
0.302 (.014)
*
MC-NNM
0.454 (.023)
0.226 (.011)
0.159 (.008)
0.140 (.007)
0.124 (.006)
0.009
CFRNet
0.316 (.025)
0.291 (.003)
0.143 (.008)
0.353 (.035)
0.104 (.007)
0.095 (.005)
CRN
0.307 (.022)
0.335 (.023)
0.316 (.022)
0.282 (.018)
0.563 (.035)
0.457 (.028)
RMSN
0.311 (.028)
0.334 (.027)
0.493 (.032)
0.342 (.032)
0.390 (.032)
0.557 (.036)
CGP
0.561 (.036)
0.561 (.035)
0.549 (.035)
0.578 .037)
0.602 (.038)
0.611 (.038)
INN
1.356 (.072)
1.614 (.078)
1.575 (.078)
1.322 (072)
1.384 (.083)
1.744 (.098)
Pretreatment Match. In the original synthetic control method and its variant models (e.g., robust synthetic control, SyncTTwin, etc.), synthetic control is defined as weighted average of the units in the donor pool (control group). The post-treatment values of the weighted average are regarded as the counterfactual prediction of post-treatment outcomes. However, the pre-treatment values of the weighted average are not the same as the observed pre-treatment data because the problem of finding the optimal (nonnegative) weights is basically deternnner linear system of equations linear system of equations that has no unique solution due to nonnegativity regularization, and it only finds an approximated solution.
Empirically, Figure shows that there are gaps between the black line (factual data) and the blue line (SCM prediction) in the pre-treatment periods, which suggests the infringement that the pre-treatment
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Table 5: Mean absolute errors by SSA-GMM, SyneTTwin, and SC ITE over prolonged post-treatment durations with no missing data and po = 0.5 confounding bias level. The best performer is in bold.
N No + Ni = 200 + 200 = 400
N = No + Ni = 1000 + 200 = 1200
Method
To = 25
To = 25
To = 25
To = 25
To = 25
To = 25
T 30
7 7= 35
T = 40
T 30
T= 35
T= 40
SSA-GMM Swnn
0.076(.000 0004)
0.094(.00 005) 0.122(.005)
0.069(.004
0.056(.003
0.045( 0.045(.002)
0.120 (.008)
0.118 (.009)
0.125 (.009)
0.089 (.005)
0.093 (.005)
0.092 (.005)
SC
0.150 (.018)
0.189 (.032)
0.192 (.032)
0.158 (.023)
0.142 (.020)
0.222 (.037)
predictions (blue line) are also affected reversely by the policy in 1988. Furthermore, Figure 4 of the SyncTTwin paper also shows the infringement. The authors say the synthetic twin matches its tre-treatment outcomes "closely," but our SSA-GMM results "exactly" match any mretreetment outcomes. Figure of the CGP paper (Schulam & Saria, 2017) also shows the gaps between the observed pre-treatment measurements (green points) and the model prediction (blue curves). Even a very small difference in the pre-treatment periods usually can make big difference in the post- treatment prediction, and the restriction cannot be ignored. Our contribution to the self-evident constraint is clear and not limited.
EM algorithm. When the number of mixture components is K 1, we can ensure the unique solution of target parameters under rotational and scaling ambiguity (i.e., WS = (R 'S) W'S' that gives global maximization of the likelihood function. However, Gaussian mixture model with K > is not free of local maxima issues, and EM algorithms only find local optimal solutions of the likelihood function. Our model also does not give the same unique parameters every time it is trained. Normally, however, as we are not too concerned about non-unique solutions from the conventional use of a Gaussian mixture model for density estimation, the authors do not think is a concerning issue for the generative task. Empirically, we observe that the probability of getting some worse solutions is very low.
Our method is unique algorithm distinguished from the SCM-based framework. SCM (synthetic control method) algorithms do find convex combination of the control group data to generate synthetic control. Originally, SCM does not use linear observers like our (()) and does not use a latent representation of data like our state vector representation S. It is true that SyncTTwin uses a latent representation (as shown in Figure of their paper), which makes it look similar to ours, but Sync'Twin also finds a convex combination in the latent representation space (as shown by Equation of their paper) like the SCM-based methods. Our method does not solve such a combinatorial problem to generate counterfactuals, Our method is probabilistic generative model that can randomly generate counterfactuals from the null space (spanned by w(To), to pretreatment observers, which ensures the pretreatment causality restriction.
Linear Observers.
Additive intervention. This paper emphasizes the restriction on the counterfactual generation that receiving treatment at a time cannot cause any difference reversely to pretreatment observation data. The restriction is obeyed in this paper by assuming such linear and additive intervention. Because the linear and additive intervention is made only in the null space to pretreatment observers (as shown in Assumption 4 and Definition 2.2), the intervention does not cause any effect on pretreatment observation data. If we decided to use nonlinear observers, we would have to assume nonlinear intervention to keep the restriction. In a bigger frame, we can consider two equally qualified options: one is that we assume linear observer system and employ Gaussian mixture model, and the other is that we assume a nonlinear observer system and use a single Gaussian model. This paper takes the first option and, in the first option, the assumption of linear and additive intervention is reasonable.
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fectual (ground truth) iground counterfactual lsyntheticl
factual Iground truth)
(chalesternl level))
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counteefactud (synthatic)
13
Figure 4: Results from complete dataset of N, 200 and Po 0.1. We used parameters of Ko 2
and 85. All the factual data are given from the statins treatment group.
20

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Published as a conference paper at ICLR 2024
C PROOFS
Proof of Lemma 2.7
The cross-sectional probability density function is given as
(ss E Pol u. Σ)
= N(WoE + solu1 Σ)
JEP N(s u,
JEN sol u1, I)
N(WoE 80: Σ)
NOO /V I (1 -80), W/I Wo)
(E) (W]Z (H (H-80).
=
Proof of Lemma
2.2 The crossnsectional Gaussian mixture distribution is given as
Σx V E Polle Ik)
Σg TW + so So Σn)
SAEP. En Th Th N(s) Mg: Ix) SE N Hk Σxe
En TW 5o, Σx)
En TW so. IN) / /NOO
(HW - So), W Wo)
Woo
TON(0 so. Ix) / /N(O// (M -8)), W (W,)
E
EN T so: ) /N W I (4w--8) W (Wo) Wo)
X
NWWoE 5o. Ix)
T00 So. Ix) NO W. (14 so). -
TWW(Wom; + soluu: Cx)/
Σ
EN TR. + solu Σx) Ck.1 N(Emmm C+)
where my = (W (w) _ (H4 - So) and C₁ = (W/ - Wo) by applying Lemma
to each Gaussian.
Proof of Theorem 2.3
From Definitions 2.2 and 4 we have for the n-th unit with treatment Wn
x (1)
(Sn.o,. (
will
(sm,.e.))
WION (To)) En
where En,o, is given by Lemma 2.2 Wo N म5K Tk My Me, Is E Ix
Sg (sn,w,)
Proof of Corollary 2.3.1 From Definitions 2.1 2.2 and Assumption 2 we have
( sw + n())
( + (To) g) + +n(()) w (n = =x=
where
=1, 1 To-
Proof of Corollary 2.3.2
ITE is the difference of treated outcome from untreated outcome. From
Definitions 2.2 and
(TB())
w efffer
(2w 1) + (1 2w) W') N((T)))) E))
= (1
2w wo wir (())
effect
where & a random variable from Lemma 2.2 by substituting Wo with (T)) and Waffect is explained in the main paper.
Proof of Theurem 2.4 From Theorem 2.3.2 and Eq.
(
(1 2w w T
( (TO
efieet
= (1 2w ww effecct
(1 (TT))
Σ Pax Mak:
21

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Published as a conference paper at ICLR 2024
Cov(E]
E|EE"] B|EJE[E]"
Σ (Cok + mau
L Park Part mark
Pakm )
E Pas (Cok (mak-
PoR mak!) (mok Σ
Σ Pak pos:mok))
Σ Pase (C Ch
and
( - 1
(1 2 werfee (())) Cov(E) (T6) W(OT
Weffect
w
effect
W(0
Σ
Pisk CCwiitiin
Cake
Chetwoonn
N W(T
Wellect-
22


Reviews:
[{"review_id": "FmS7yFFiXU", "reviewer": "Reviewer_njab", "summary": "This work focuses on estimating time-varying individualized treatment effects within a linear model. The authors formulate a linear static-state model and provide a novel method to estimate the corresponding counterfactual data. They have demonstrated that the proposed approach can achieve better performance in their experiments.", "questions": "Can you provide an intuitive explanation for Assumption 2? Typically, people use a linear dynamical system model to describe the generative mechanism. It would be beneficial to include some descriptions in the main text rather than placing them in the appendix.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Generating the counterfactual data is a challenging task.\n\n2. The proposed method is both novel and interesting.\n\n3. The analyses are sound and presented in a logical manner.", "weaknesses": "1. The method is only applicable to linear models.\n\n2. In some instances, the description is not sufficiently clear. For example, the model's explanation here lacks clarity, particularly in the description of assumption 2, which necessitates further comprehension through reading the instructions in the appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mUfIUqo0Pe", "reviewer": "Reviewer_EGeA", "summary": "This paper focuses on counterfactual generation. In particular, it introduces a Gaussian mixture model as a prior probability distribution over the static states, and uses it as a generative model for synthetic counterfactual longitudinal data. The proposed method can be used to estimate the individual treatment effects (ITE) using the synthetic counterfactual outcomes data.", "questions": "Are the reported results regarding ITE or the expected value of ITE?", "limitations": "", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "This paper considers counterfactual generation, which is an important problem. Also, it uses a Mixture of Gaussians to approximate general data distributions; this is the main difference compared to previous papers.", "weaknesses": "The difference of the proposed method is that it assumes a Gaussian mixture model to approximate distributions, in contrast to previous works which employed neural networks. However, the experimental results may not provide sufficient evidence to fully assess the performance of the proposed method. Notably, the paper lacks experimental results on benchmark datasets such as IHDP and Jobs.\n\nThe paper asserts its ability to identify ITE as defined in Equation 11 and provides results on ITE estimation accuracy. However, it's important to note that the defined ITE is a random variable. Consequently, I'm uncertain whether the reported results pertain to ITE itself or its expected value.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4GMa2dqcHw", "reviewer": "Reviewer_TJyJ", "summary": "The paper proposes a linear algebraic framework for generating synthetic counterfactual data that matches pretreatment factual data. The approach claims to be the first-ever counterfactual generative model for creating personalized clinical trial digital twins. The method outperforms other methods of counterfactual generation and individual treatment effect estimation, as demonstrated using simulated ground truth counterfactual data. Additionally, the paper provides a formula to estimate the time-varying variance of individual treatment effects, which can be interpreted as a confidence measure for the generated counterfactuals.", "questions": "Regarding Fig.4, what methodology did the authors employ to randomly produce counterfactual pre-treatment data?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-written and easy to follow.\n\n- The experiments on California cigarette sales data and simulated LDL cholesterol data showcase the effectiveness of the proposed method as some extent.", "weaknesses": "- The motivation of this paper is unclear. The contribution also seems limited. The restriction on counterfactual generation that receiving treatment at a time $T_0$ cannot cause any difference reversely to counterfactual generation at $t < T_0$, comes across as self-evident. It remains ambiguous whether prior research infringed upon this constraint. The comparative advantages of the introduced method with the existing work are not distinctly highlighted, leading to a gap in understanding its unique value proposition.\n\n- To evaluate how true synthetic counterfactuals are, more experiments on synthetic datasets including ground-truth counterfactual data are required. It would be insightful to demonstrate the method's robustness under diverse treatment times and over prolonged post-treatment durations. \n\n- The paper could benefit from a rigorous theoretical exploration addressing the method's resilience against noise and confounding bias. Would this be challenging with such a linear framework?\n\n- There is a lack of literature review or an introduction to some of the methods appearing in the experimental part, such as MC-NNM (Athey et al., 2021) and CGP (Schulam and Saria, 2017).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1gsKnNRJs5", "reviewer": "Reviewer_aGd4", "summary": "This paper investigates the problem of generating synthetic counterfactual data that exactly matches pretreatment factual data under a linear algebraic framework (namely, the static state of the factual and counterfactual data are all in affine subspace). By defining the shift from a factual state to a counterfactual state under a soft intervention, the counterfactual data can be generated by solving the log-likelihood function as a static state analysis with priors of an ω-dependent Gaussian mixture model. Such a generation method can be used to estimate an ITE and the experimental results of the simulated data show that the proposed method outperforms the baselines in estimating ITE.", "questions": "Refer to \"Weaknesses\".", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper deals with a challenging but important problem of counterfactual generation, which may be useful to the field of causal inference.\n\n2. A practical estimating method is provided in estimating parameters (EM algorithms) and formalizes a generation process of counterfactual data according to estimated results. \n\n3. The experimental results verify the generated counterfactual data is useful for estimating ITE.", "weaknesses": "1. My main concern is the identification of parameters in the log-likelihood function, i.e., how to ensure the unique solution of target parameters by the EM algorithm. Because this is a key issue in generating the true counterfactual data.\n\n2. I think it is necessary to discuss the reasonableness for the same linear observation \"W\" in the factual data and counterfactual data generation. Because if we resolve the \"W\" from factual data, we can naturally obtain the counterfactual state under this setting. Such a method does not surprise me and I guess there are some similarities with the SCM-based framework (used for estimating counterfactual framework).\n\n3. Assumption 4 assumes that the soft intervertion on \"s\" is linearity and additive. Some limitations or reasonableness to it also require to be discussed.\n\nBesides, as this is a bit out of my expertise, my confidence on the score is on the lower spectrum.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)