Estimating Treatment Effects using Neurosymbolic Program Synthesis
Authors: Abbavaram Gowtham Reddy
Vineeth N. Balasubramanian
Venue: Submitted to ICLR 2023 (2023)
Keywords: Causal effect
treatment effect
neurosymbolic programming
domain specific language

Abstract: Estimating treatment effects from observational data is a central problem in causal inference. Methods to solve this problem exploit inductive biases and heuristics from causal inference to design multi-head neural network architectures and regularizers. In this work, we propose to use neurosymbolic program synthesis, a data-efficient, and interpretable technique, to solve the treatment effect estimation problem. We theoretically show that neurosymbolic programming can solve the treatment effect estimation problem. By designing a Domain Specific Language (DSL) for treatment effect estimation based on the inductive biases used in literature, we argue that neurosymbolic programming is a better alternative to treatment effect estimation than traditional models. Our empirical study reveals that our model, which implicitly encodes inductive biases in a DSL, achieves better performance on benchmark datasets than the state-of-the-art models.

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ESTIMATING TREATMENT EFFECTS USING NEUROSYMBOLIC PROGRAM SYNTHESIS
Anonymous authors Paper under double-bling review
ABSTRACT
Estimating treatment effects from observational data is central problem in causal inference. Methods to solve this problem exploit inductive biases and heuristics from causal inference to design multi- heatd neural network architectures and reg- ularizers. In this work, we propose to use neurosymbolic program synthesis, a data-efficient, and interpretable technique, to solve the treatment effect estimation problem. We theoretically show that neurosymbolic programming can solve the treatment effect estimation problem. By designing Domain Specific Language (DSL) for treatment effect estimation problem based on the inductive biases used in literature, we argue that neurosymbolie programming is a better alternative treatment effect estimation than traditional methods. Our empirical study reveals that our method, which implicitly encodes inductive biases in a DSL, achieves better performance on benchmark datasets than the state-of-the-art methods.
INTRODUCTION
Treatment effect (also referred to as causal effect) estimation estimates the effect of a treatment variable outcome variable (c.g.. the effect of drug recovery). Randomized Controlled Trials (RCT's) are widely considered as the gold standard approach for treatment effect estimation (Chalmers) et al. 1981 Pearl 2009).. In RCTs, individuals are randomly split into the treated group and the control (untreated) group. This random split removes the spurious correlation between treatment and outcome variables before the experiment
that estimated treatment effect is unbiased. However, RCTs are often: (i) unethical (e.g., in study to find the effect of smoking on lung disease, a randomly chosen person cannot be forced to smoke), and/or (ii) impossible/iffesiblee (e.g.. in finding the effect of blood pressure on the risk of an adverse cardiac event, it is impossible to intervene on the same patient with and without high blood pressure with all other parameters the same) (Sanson-Fisser et al. 2007 Carey & Stiles 2016: Pearl et al., 2016). These limitations leave us with observational data to compute treatment effects.
Observational data, similar to RCTs, suffers from the fandamental problem of causal inference (Pearl 2009). which states that for any individual, we cannot observe all potential outcomes at the same time (e.g., once record a person's 's medical condition after taking a medicinal drug, cannot observe the same person's ' medical condition with an alternate placebo). Observational data also suffers from selection bias (e.g., certain age groups more likely to take certain kinds of medication compared to other age groups) (Collier & Mahoney 1996). For these reasons, estimating unbiased treatment effects from observational data can be challenging (Hernan & Robins 2019 Farajtabar et al., (2020).
However, due to the many use cases in the real-world, estimating treatment effects from observational
data is one of the long-standing central problems causal inference (Rosenbaum & Rubin 1983
1985
Brady et al. 2008 Morgan & Winship 2014 Shalit et al. 2017 Yoon et al. 2018 Shi et al.
2019
Yao al. 2018 Zhang 2021).
Earlier methods that estimate treatment effects from observational data are based on matching techniques that compare data points from treatment and control groups that are similar w.r.t.
metric (e.g., Euclidean distance in nearest-neighbor matching, or propensity score in propensity score matching) (Brady et al.) 22008) Morgan & Winship, 2014). Recent methods exploit induc- tive biases and heuristics from causal inference to design multi-head neural network (NN) mod-
els and regularizers (Hill) 2011 Farajtabar et al.] 2020; Shi et al.] 2019 Schwab et al. 2020
Chu al.] 2020; Shalit et al.] 2017: Alaa & van der Schaar 2017; Yoon al. 2018 Bica
et al. 2020 Künzel et al.] 2019; Chermozhukov et al.) 2018; Yao et al.] 2018: Zhang et al.
2021). Multi-head NN models are typically used when treatment variables single-dimensional

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and categorical (Shi et al. 2019 Shalit al. 2017; Farajtabar et al. 2020 Schwab et al.
2020), and regularizers therein enforce constraints such as controlling for propensity score in- stead of pre-treatment covariates, i.c. covariates that are not caused by the treatment variable in the underlying causal data generating graph (Shi et al., 2019: Rosenbaum & Rubin 1983). However, each such model is well-suited to cer-
CFRNe TARNE
tain kind of causal graph, and may not apply to
all causal data generating processes. For example,
(0), TO
Existing Methaols
as shown in Fig D CFRNet (Shalit et al. 2017). a popular NN-based treatment estimation model, controls mretrreateett covariates using regular- izer based on an Integral Probability Metric (IPM). It requires the representations of non-treatment covariates (denoted as x in Fig I with and with- out treatment to be similar. This is relevant for causal model A in the figure, but does not work for causal model B, where tre-tmetttent covari- ates are caused by the treatment / and hence could vary for different values of One would ideally need a different regularizer or architecture to ad- dress causal model B (the same observation holds for TARNet (Shalit et al. 2017) too). In practice, one may not be aware of the underlying causal model, making this more challenging. In this work, we instead propose to use a neurosymbolic program synthesis technique to compute treatment effect, which does not require such explicit reg- ularizers or architecture redesign for each causal model. Such a technique learns to automatically synthesize differentiable programs that satisfy a given set of input-output examples (Shah et al.) 2020 Parisotto et al.) 2016), and can hence learn the sequence of operations to estimate treatment effect for this set. We call our method as the NEuroSybbolic Treatment Effect EstimatoR (or NESTER).
CPM (96"
Can Model
Can Not Mode X
Censal Models
Σ Flythelpto
Rlgsknn9) Mare)
Can Model
Can Model /
(subser [t,, x), [0.]]))
Proposed NESTER then transform ([t, (t.))))
subset (t.x), [0.1])
transfor(((xx) P..
Figure I: IPM regularization in CFRNet (note that CFRNet is combination of a simple two-hcad TAR- Net with IPM regularization) controls for covariates ar which may lead to incorrect treatment effect w.r.t. causal model B. However. NESTER learns synthe- size different estimators for the two causal models.
Neurosymbolie program synthesis is known have the flexibility to synthesize different programs for different data distributions to optimize a performance criterion, while still abiding by the inductive biases studied in treatment effect estimation literature (see Sec 4.1 4.1 for more details). To describe further, one could view CFRNeU/TARNet as implementing one if then - else program primitive with its two-headed NN architecture. NESTER will instead automatically synthesize the sequence of program primitives (from a Noomain-speciff language of primitives) for given set of observations from a causal model, and can thus generalize to different distributions.
Program synthesis methods, in general, enumerate set of programs and select (from the enumer- ation) a set of feasible programs that satisfy given input-output examples so that the synthesized programs generalize well to unseen inputs (see Appendix for example) (Biermann) 1978 Gulwani 2011 Parisotto et al., 2016 Valkov et al., 2018 Shah et al. 2020). Usually, a Domain- Specific Language (DSL) (e.g.. specilic context-free grammar) is used to synthesize relevant programs for a given domain and task. Recently, various NN-based techniques have been proposed to perform neurosymbolic program synthesis (Parisotto et al. 2016 Valkov et al.] 2018 Gaunt et al. 2017] Bošnjak et al. 2017). We use the neurosymbolic program synthesis paradigm where each program primitive (e.g.. if then else, 01 + (22) is a differentiable module (Parisotto et al.) 2016 Shah et al. 2020). Such differentiable programs simultancously optimize program primitive parameters while learning the overall program structure and flow. The set of possible programs that can be synthesized using DSL is often large (Parisotto et al. 2016). Many methods have been proposed to search through the vast search space of programs efficiently (Gulwani et al.) 2012 Parisotto et al. 2016, Valkov et al., 2018 Shah et al.) 2020). We use Neural Admissible Relaxation (Shah et al. 2020) in this work, which uses neural networks as relaxations of partial programs while searching the program space using informed search algorithms such as A" (Hart et al. 1968). The final program can be obtained by training using gradient descent algorithms. Our key contributions are:
We study the use of neurosymbolic program synthesis as a practical approach for solving treatment effect estimation problems. To the best of our knowledge, this is the first such effort that applies neurosymbolic program synthesis to estimate treatment effects.
We propose a Domain- Sppecffic Language (DSL) for treatment effect estimation, where each program primitive is motivated from basic building blocks of models for treatment effect estimation in literature.
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We theoretically show that our neurosymbolic program synthesis approach can approximate continuous function upto an arbitrary precision. This result enables us to solve the treatment effect estimation problem by assuming a continuous function relating the treatment and outcome variables.
We perform comprehensive empirical studies on multiple benchmark datasets (including additional results in the Appendix) where we outperform existing saute-o--the-ane models. We also show the interpretability of such a neurosymbolic approach on synthetic as well as real-world datasets, thus highlighting the usefulness of our approach over traditional treatment effect estimation methods.
2 RELATED WORK
Traditional Methods for Treatment Effect Estimation. Early methods of treatment effect estimation from observational data are largely based on matching techniques (Brady et al. 2008 Morgan & Winship 2014; [tuart 2010) where similar data points in treatment and control groups are compared using methods such as nearest neighbor matching and propensity score matching to estimate treatment effects. In nearest neighbor matching (Stuart) 2010), for each data point in the treatment group, the nearest points from the control group w.r.t. Euclidean distance are identified, and the difference in potential outcomes between treatment and corresponding control data points is estimated as the treatment effect. In propensity score matching (Rosenbaum & Rubin 1983), a model is trained to predict the treatment effect value using all data points from both treatment and control groups. Using this model, points from treatment and control groups that are close w.r.t. the model's output are compared, and the difference in potential outcomes of these points is estimated as treatment effect. However, such matching techniques are known to not scale to high-dimensional or large-scale data (Abadie & Imbens, 2006).
Another family of methods estimates treatment effects using the idea of backdoor adjustment (Pearl 2009; Rubin 2005). Under the assumption of availability of a sufficient adjustment set (Pearl 2009). these models rely on fitting conditional probabilities given the treatment variable and sufficient adjustment set of covariates. However, such models are known to suffer from high variance in the estimated treatment effects (Shalit et al., 2017). Covariate balancing is another technique to control for the confounding bias to estimate treatment effects. Weighting techniques perform covariate balancing by assigning weights to each instance based on various techniques (e-g.. weighting each instance using propensity score in the inverse probability weighting technique) (Rosenbaum & Rubin
1983 Assaad et al., 2021] CRUMP et al. 2009) L & T 2013 Diamond & Sekhon 2013 Li & Fu
2017). As noted in (Assaad et al. 2021). ) such methods face challenges with large weights and
high-dimensional inputs. Besides, leveraging the success of learning- basen methods has delivered
significantly better performance in recent years.
Gearning-based Methods for Treatment Effect Estimation. Recent methods to estimate treatment effects have largely been based on multi-headed NN models equipped with regularizers (Hill 2011
Farajtabar et al. 2020 (hi et al.) [019 Schwwb et al., [220]] Chu et al.] 2020; (halit et al. 2017
Yoon et al. 2018; Bica et al. 2020). To find treatment effects under multiple treatment values and continuous dosage for each treatment, (Schwab al. 2020) devised an NN architecture with multiple heads for multiple treatments, and multiple sub-heads from each of the treatment-speciffc heads to model (discretized) dosage values. CFRNet (Shalit et al. 2017) proposed two-headed NN architecture with a regularizer that forced representations of treatment and control groups to be close to each other, in order to adjust for confounding features before forwarding the representation to -reatfent-specffic heads. Extending CFRNet architecture, (Farajtabar et al., (0200) proposed an additional regularizer to adjust for confounding by forcing both rreatment-specife heads to have same baseline outcomes (i.e., for any data point, both treatment-specifu heads should output same value). Dragonnet (Shi et al. (019). along with two heads for predicting treatment-specine (potential) outcomes, additional head to predict treatment value was also used; this allowed pre- treatment covariates to be used in predicting potential outcomes. Assuming that potential outcomes strongly related, (Curth & van der Schaar 202T) proposed techniques that improve existing models using the structural similarities between potential outcomes. All of these methods, however, have fixed architecture design and can hence address observational data from certain causal models. Our approach is also NN-based but uses a neurosymbolic approach to automatically synthesize an architecture (or a flow of program primitives), thereby providing it a capability work across observational data from different causal models conveniently.
Generative Adversarial Networks (GANs) (Goodfellow et al. 2014) have also been used to learn the interventional distribution (Yoon et al. 2018 Bica et al. 2020 from observed data in both
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categorical and continuous treatment variable settings estimate treatment effects. By disentangling confounding variables from instrumental variables, (Zhang et al.] (021)) proposed a variational inference method for treatment effect estimation that uses only confounding variables. However, generative modeling requires a large amount of data to be useful, which is often not practical in treatment effect estimation tasks. (Yao al.] 2018) proposed a method to learn representations by leveraging local similarities and thereby estimate treatment effect, Ensemble models such as causal forests (Wager & Athey 2018), and Bayesian additive regression trees (Chipman et al. (010) have also been considered for interval estimation. As stated earlier, our work is however very different from these efforts, and seeks to build a flexible yet powerful framework for treatment effect estimation using neurosymbolic program synthesis.
Neurosymbolic Program Synthesis. Program synthesis, viz. automatically learning a program that
satisfies a given set of input-ouiput examples (Biermann) 1978 Gulwani 2011 Parisotto et al. 2016 Valkov et al. 2018 Shah et al., (2020), has been shown to be helpful in diverse tasks such as low- level bit manipulation code (Solar-Lezama et al. 2005), data structure manipulations (Solar Lezama 2008 and regular expression-based string generation (Gulwani 2011). For each task, specific DSL i8 used to synthesize programs. Even with a small DSL., the number of programs that can be synthesized is very large. Several techniques such greedy enumeration, Monte Carlo sampling, Monte Carlo tree search, evolutionary algorithms, and recently, node pruning with neural admissible relaxation have been proposed to efficiently search for optimal programs from vast search space (Gulwani et al., et al. 2016 Valkov et al. 20188; Shah et al., (220). We use the idea of node pruning with neural admissible relaxation (Shah et al.) (020) in this work as it gives near-optimal solutions with fast convergence. This is the first use of neurosymbolic program synthesis for treatment effect estimation, to the best of our knowledge.
BACKGROUND AND PROBLEM FORMULATION
Treatment Effect Estimation: Let D {xx, L4, y:) Hi= be an observational dataset of n triplets.
Each triplet (x,, 44s 31) is sample drawn from the true data distribution p(X, T, Y), where X, Y and T are the corresponding random variables (described herein). x; E R denotes the d- dimensional covariate vector, & E E denotes the treatment value (t; is not part of x,), and ys E R denotes the corresponding outcome. To explain treatment effects, consider a simple setting where treatment binary-valued i.e., f E (0, 1). For the 2th observation, let Y?O denote the true potential outcome under treatment 42 = 0 and Y? denote the true potential outcome under treatment 6; = 1. Because of the fundamental problem of causal inference, we observe only one of Y, Y! for given x.]. The observed outcome y/ can be expressed in terms of true potential outcomes as: y/: = 4, Y, + (1-(4)) Y, One of the goals in treatment effect estimation from observational data is to learn the estimator fxx, t) such that the difference between estimated potential outcomes (i.e., under t t= 1 and = 0): f(x;; 1) - [(x:, 0) is close as possible to the difference in true potential outcomes: Y y Vi. This difference for specific individual is known as Individual Treatment Effect (ITE) (Pearl 2009). Extending the discussion on ITE an entire population, our goal is to estimate the Average Treatment Effect (ATE) of the treatment variable I' on the outcome variable Y which is defined as:
ATEX Eoo 1)] E|y [do 0)]
where the do(.) notation denotes external intervention to the treatment variable (Pearl 2009). i.e.
E[Y]do(T = [)] refers to the expected value of the outcome Y when every individual in the population is administered with the treatment 4. (Note that treatment is not binary-valued, treatment effects are calculated w.r.t. a baseline treatment value (Pearl 2009). and the right term in Eqn [ would compute the interventional expectation at the baseline.) ) Assuming X satisfies the backdoor criterion relative to the treatment effect of T on Y (Pearl 2009), we can write |do(T t)] Ex-x E E = t, X = x]]. Using this, a simple technique to estimate E[Y[T = t, = x] (and thus E|y]do = t)}, the ATE) is to fit model for Y given T, and X. These models are the basic building blocks of most methods for treatment effect estimation. We use the finite sample approximation of ATE by taking the average of ITEs. Following (Shalit et al. 2017 Lechner [001]
Imbens| 2000; Schwab et al. 20200 Zhang et al. 2021). 1, we make the following assumptions which
are sufficient to guarantee the identifiability (Pear 2009) of treatment effects from observational data.
This is also referred to as no unmeasured confounding assumption. For a given set of pre-treatment covariates, treatment is randomly assigned. Mathematically, in binary treatment setting, conditioned a set of pre- treatrettmntt covariates X, treatment is indepedent of the outcomes yo yl (i.e., (Y", Y1) 1. X).
Ignorability:
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Positivity: Treatment assignment for each individual is not deterministic, and it must be possible to assign all treatment values to each individual, i.e. 0 < p(t|x) < Vt, x.
Stable Unit Treatment Value Assumption (SUTVA): The observed outcome of any individual under treatment must be independent of the treatment assignment to other individuals.
Neurosymbolic Program Synthesis: Following (Shah et al. 2020), let (P,0) 0) be a neurosymbolic program where P denotes the program structure and / denotes the program parameters. (P, 0) is differentiable in (see Appendix for an example neurosymbolie program). P is synthesized using a Context- Free Grammar (CFG). A CFG consists of a set of rules of the form a non-terminal and 01.-- Tn are either non-terminals or terminals. Program synthesis starts with an initial non-terminal, then iteratively applies the CFG rules produce series of partial structures, viz. structures made from one or more non-terminals and zero or more terminals. These partial structures are considered as nodes in program graph. The process continues until no non-terminals are left, i.e., we have synthesized a program. The leaf nodes of the resultant program graph contain structures that consist of only terminals. Let be the set of parameters of such a leaf node structure P. Let s(r) be the cost incurred in using the rule r while generating program structure. The structural cost of P is s (P) = Exre(()) s(r), where R(P) is the set of rules used to create the structure P. In this paper, we set s(r) to a constant real number for all production rules (e.g.. s(r) = Vr E R(P)). The program learning problem is thus usually formulated as a graph search problem, i.e., starting with an empty graph, the graph is expanded by creating new partial structures (internal nodes of the graph) and structures (leaf nodes of the graph). When searching for an optimal program, parameters of the program (and program structures) are updated simultancously along with the synthesis of the programs (Shah et al. 2020).
0i...., On where
For a synthesized program (P. we define ((P,0) = -(x,,,)>D((C (x, f) -) as the error/cost incurred by (P, 0) in estimating treatment effects. The overall goal of neurosymbolic program synthesis is then to find a structurally simple program (that can also be human-interppettbble) with low prediction error, i.e. to solve the optimization problem: 0)). We now describe our methodology.
= arg wigmin (r (p (s(P)
4 NEUROSYMBOLIC TREATMENT EFFECT ESTIMATOR: METHODOLOGY
The overall idea of our methodology is to design a Somain-Speciife Language (DSL) for treatment effect estimation that is fairly general, followed by the use of the standard A" search algorithm to synthesize programs given observational data from a specific causal model. We begin by discussing the DSL we design, followed by the program synthesizer. Note that one could view each primitive of our DSL as modules of existing learning- based treatment effect estimators such as TARNet or CFRNet (Shalit et al. 2017). We also theoretically analyze the usefulness of the search-based neurosymbolic program synthesizer for the given task.
4.1 DOMAIN SPECIFIC LANGUAGES FOR TREATMENT EFFECT ESTIMATION
Since program synthesizer requires input set of input-output examples, unsurpris- ingly, we can pose the problem of treatment effect estimation as the problem of map- ping a set of inputs to corresponding outputs. Concretely, given observational data D, the set of pairs (1) act as inputs and the set of outcomes { act as outputs.
For simplicity, let v; = xt] (concatena- tion of treatment and covariates) denote the 7th input. A synthesized program learns to estimate the potential outcomes for un- seen inputs by learning a mapping between given input-output examples. To bring in- terpretability to synthesized programs and to leverage the inductive biases considered in treatment effect estimation literature, we develop DSL (Table ( based on well- known program primitives that have con- nections to ideas used in literature for treat- ment effect estimation (illustrated in Ta- (We later state Propn 4.2 that guar- antees the existence of a DSL for treatment effect estimation task). As discussed earlier, existing treatment effect estimation methods introduce inductive biases into machine learning models either
Regularizen/ Architectural Changes
Program Synthesis Alterative
Twoohhedd/Muli-heeee network
1f then else subset
(Tarajtabur el al. 2020 (Shi et al.,
2019
(hhlit el al. 2011 Schwab el al.
2020
Pre-treatment selection.
subeet
Propensity Score Matching Shi ct
2019
regularization
transform
Shalit et al. 2017
Parajtubar al. 2020
Table 1:
Connection between inductive biases in existing literature and the program primitives in the proposed DSL. This equivalence allows us to device a method that doesn't t require additional regularizers or architectural changes.
ble II)
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A DSL for the Treatment Effect Estimation Task
2m if then a else a transform(e; H, o) subset(a,(a.B)) const(() o (a,a))
Program Primitive
Description
if then a else a
Simple then else condition. To avoid evaluating conditions, and to enable back- propagation, we implement smooth approximation of if then else.
2. transform(a, 14, o)
Transform the input vector e into p(a) as ó(a) = 2- where jf and a are mean and standard deviation of observational data. Feed ((a) into NN to get a real number as output.
3. subset(a, (a..b)
Select a set of features from the start index (including) to end index b (excluding) from the input o. Other features are set to O. Feed this vector into NN to get a real number as output.
4. const(8)
Learn set of constants of shape S.
5. o(a,c) a))
Parameterized algebraic functions (e.g.. Or + 0g, 0r 02; A2 R).
Table 2: DSL. for the treatment effect estimation task in Backus-Naur form [Winskel 1993) and its semantics.
v represents input from D) More details of each primitives is provided below.
through regularizers or through changes in NN architectures. On a similar note, one could view a DSL set of inductive biases based on learnable program primitives (Shah al. 2020 Chaudhuri et al. 202T). The proposed DSL is based on inductive biases used in treatment effect estimation literature. We next describe the connections between program primitives in our DSL in Table 2 and inductive biases used in traditional treatment effect estimation methods.
Connection between multi-head neural network architectures and if - then - else, subset: Recall that, in treatment effect estimation, our goal is to estimate the quantity E| = X = x]. If a single model is used to estimate both E (T' = 0, X x] and E the case that X is very high dimensional and hence the treatment T, which is often one dimensional, may be discarded by the model when making predictions. This will result in the estimated treatment effect being biased towards (Künzel et al. 2019). To account for this, two separate models can be used to estimate E|y T 0, X x E 1,X x]. However, this method suffers from high variance in estimated treatment effect due to limited data treatment-speciffe sub-groups and selection bias (Shalit et al. 2017).
X = x], it is often
In order to mitigate this problem. (Shalit et al. 2017) and subsequent efforts by (Shi et al. 2019) Schwab al. 2020 Farajtabar et al. 2020) an NN architecture in which two separate heads are spanned from a latent representation layer to predict treatment specific outcomes and thus achieve better treatment effect estimate with lower variance. Such two-head NNs can be implemented using a combination of if - then else and subset program primitives. For example, to implement two-head NN architecture, neurosymbolic program synthesizer can perform the following: if 0n = subset(y (vo [0). = 1 (subset(y. (v, ba..b)) takes a vector v as input and returns a real number as output as explained later in this section), the program synthesizer executes ag else it executes ag where 022, ag are two different sub-structures that act as two heads of the overall architecture
Note that each in the primitive: "if a then a else a" returns a real number and hence the output of "if a then a else a" is also a real number. For e.g., as discussed above, a here can be either subset(v, [)..1) or transforml (v, μ,, too. Here both subset(v, [0..1]) and transf sansorrm(yy. 14, o) take vector v as input and return a real number as output.
To avoid discontinuities and to enable backpropagation, following (Shah et al., 2020), we im- plement a smooth approximation of if then else. For example, smooth approximation of if a >>> then else c can be written as (B3. a) b (1 a(3. a)) c, where a is the sigmoid function and / is temperature parameter. As 8 + 0, the approximation approaches the usual - then else. It is now easy to see that multi-head NN architectures can be implemented using multiple if then else and subset primitives. It is important to note that we do not hard-code/pre- www.-codd/pre-effiieee define the network architecture. Instead, the program synthesizer learns to generate programs such that the primitives are composed in any order deems be effective in minimizing the loss value during training (see Sec.5Jand Appendix for examples).
Connection between IPM regularization and tr aransform To improve the results from two head NN architectures (e.g.. TARNet), CFRNet (Shalit et al.) 2017) proposes to use IPM regularization (e.g.. Maximum Mean Diserepancy (Gretton et al.) 2012), Wasserstein distance (Cuturi & Doucet, 2014) on a latent layer representation. This enforces the encoded distribution of treatment (p(x]( = 1)) and

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control (p(x)t 0)) groups to be close to each other. Minimizing IPM between p()(x)]tt = 1) and p(o(x)]t= 0) (where á is the learned representation) is then the same as ensuring that treatment and covariates are independent (i.e., T I X) thus mimicking RCTs (Shalit et al.] 2017). To introduce this kind of inductive bias, we introduce a program primitive called transform(e) 0 H4, σ)) that transforms a given input vector a into o(a) using two other vectors J4, a as ((a)= = o-P, where ju, a are mean and standard deviations of the observational data D.
If transform(a, 14, o) is applied to the entire dataset D, the transformed data now has mean 0 and standard deviation 1 (where 0 and 1 are vectors of Os and 1s respectively). When the two subpopulations p(x|( = 0) and p(x[( = 1) are distributed similarly to 'D (which would also satisfy the ignorability assumption), the means and standard deviations of p(ó(x)]t
0) and p(o(x)]t = 1) will also be approximately equal to 0 and 1 respectively (note that 6(x) x_", and the input to transform(c. (a, [4, () is the vector [t, x] E D). Then, the Maximum Mean Discrepancy between p()(x)] it 0) and p(o(x)]t = 1) will go towards zero when matching the first two moments of p( = 0) and p( = 1) for treatment effect estimation. The transformed vector 6(x) is subsequently fed into multi-layer perceptron to produce real number as output, which is then fed into other program primitives.
transform(a. (a, f4. a) appears to be similar to data standardization, a data pre-pr rrepprcessiig step. How- ever, unlike the fixed architecture in traditional NNs. program synthesis has the flexibility to choose when to use transf crmeform((() (x, μl, or) (for e.g, x could be the output of the 3 program primitive, not input). Besides, though the reason to introduce transform(c, (4, σ) is to mimic IPM regularization (specifically Maximum Mean Discrepancy). it is evident from our DSL that the program synthesizer can use transf connforr(ee (a, μl, o))) multiple times in program (see Appendix for examples).
Connection between pre-treatment covariate selection and subset: Under the ignorability as- sumption, pre-treatment covariates are controlled to find estimates of treatment effects. For e.g.. (Shi ctt al. 2019) controls pre-treatment covariates via controlling propensity score (Rosenbaum & Rubin 1983). However, it is not required to control all the covariates in the input. In order to identify a minimal set of pre-treatment covariates to control, we subset(a, [a..b]) primitive. If we do not know which indices to select, multiple instances of subset (a..b)) can be used by assigning dif- ferent values to a, b in each instance. Program synthesizer then selects appropriate subset(a, (a..b)) for some a, b. subset (a, ba. also helps to identify the most important dimensions from a given input/hidden representation as explained in the if then- else example earlier. Finally, the chosen vector is fed into a multi layer perceptron to produce a real number as output which will subsequently be used by other primitives.
The other two simple program primitives-const(S @(a, a whhoe semantics are given in Table 2 are included for giving additional flexibility to the program synthesizer and combining various program primitives effectively to achieve better results. O(a, a) takes two real numbers as inputs and returns a real number as output after performing the algebric operation @. Using the proposed DSL, we now present the algorithm to synthesize neurosymbolic programs that estimate treatment effects.
4.2 NEUROSYMBOLIC PROGRAM SYNTHESIS FOR TREATMENT EFFECT ESTIMATION
We use the A informed search algorithm (Hart et al. 1968) to implement the proposed NESTER method. The heuristic function h we use in our method is defined as follows. For any partial structure P(u) node u, NNs with adequate capacity (enough width and depth) are used to replace the non-terminals. The training loss of the resultant program (1) 0())) on D then acts as the heuristic value hu) at the node u (Shah et al. 2020). Using this heuristic function, we run A" algorithm to find the programs that estimate treatment effects. We outline our approach in Algorithm [ We now study the theoretical guarantees of neurosymbolic program synthesis in estimating treatment effects.
Definition 4.1. (Admissible Heuristics (arrris] 1974; Pearl 1984) In an informed search algorithm. heuristic function h(u) that estimates the cost to reach goal node from node u is said to be admissible if h((w) < h1" (u), Yu where he (u) is the actual/true cost to reach the goal node from u. hu)) is said to be addmisiible if h(u) S h² h²())) + e, Vu.
Proposition 4.1. In an informed search algorithm, let the cost of the leaf edge (wi, u) (edge connecting internal node u, to leaf node u)) be s(r) + ((P, 8*), where 8" = arg ming {(P, 8) and r is the rule used to create in from us NNs N parameterized by their capacity (architecture width and height) are used to substitute the terminlls in the partial structure of u, the resultant program' 's training loss is equal to the e e-admissible heuristic value at the node ui- Such an admissible
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Under review as a conference paper at ICLR 2023
heuristic returns solution whose path cost is at most an additive constant away from the path cost of the optimal solution (Shahh et al. (020).
Proposition 4.2. Given e-admissible heuristic, for any trained I-hidden layer NN N with m imputs, n hidden neurons, and one output, there exist a Domain Specific Language C such that the error/loss incurred by the synthesized program (P,0) 0)) is close to the error/loss incurred by N in approximating any continuous function.
Proof' of the above propositions are in the Appendix. The universal approximation theorem (Homik et al. 1989 states that we can increase the number of hid- den layer neurons of a 1-hidden layer NN N to ap- proximate any continuous function ʃ with certain error, say é. Proposition 4.2 states that there exists a neurosymbolic program (P. whose error in approx- imating N/ is e. Equivalently, there exists a neurosym- bolic program (P,0) () whose error in approximating f is (c+ 2). That is, if the relationship between treatment and effect is a continuous function, neu- rosymbolic programming is viable candidate for estimating treatment effects.
Algorithm 1: NESTER using A'
Input: Source node uo. Q = {uo [(o) == 00 while Q * 0 do
V + arg min f(()
Q ( if e is goal node then
return v end
else
create new partial architectures from (children of v) using DSL L
foreach child u of do
h(u) mine(w) C(P(u), (())
f(u) <((()) + +h(u) (u(
EXPERIMENTS AND RESULTS
end
end
We perform experiments to showcase the usefulness of NESTER in estimating treatment effects when coupled with our proposed DSL. Our code along with instructions for reproducibility of results is in the supplementary material. To permit interpretability. we limit the program depth to utmost 5 for the main experiments (see Appendix for experiments with other depths).
end
Datasets and Baselines: Evaluating treatment effect estimation methods requires all potential outcomes to be available, which is impossible due to the fundamental problem of causal inference.
Thus, following (Shalit et al. 2017 Yoon et al. 2018: Shi et al., 2019 Farajtabar al. (020). we experiment on two semi semi-synthetie datasets Twins (Almond et al., 2005), IHDP (Hill [001] that are derived from real-world RCTs (see Appendix for details). For these two datasets, ground truth potential outcomes (a.k.a. counterfactual outcomes) are synthesized and available, and hence can be used to study the effectiveness of models in predicting potential outcomes. We also experiment on one real-world dataset-Jobs Londe -whee we observe only one potential outcome. Each dataset is split 64/16/20% into train/validation/test sets, similar to carlier efforts.
We compare NESTER with Ordinary Least Squares with treatment a feature (OLS-1), OLS with two regressors for two treatments (OLS-2), k-Nearest Neighbors (k-NN), balancing linear regression (BLR) (Johansson et al. 2016). Bayesian additive regression trees (BART) (Chipman
ct al. 2010). random forest (Breiman (001). causal forest (Wager & Athey 2018), balancing neural
network (BNN) (Johansson et al. (016)), treatment-agnostic representation network (TARNet) (Shalit jet al. 2017), multi-head network (MHNET) (Farajtabar et al., 2020), Generative Adversarial Nets for inference of individualized treatment effects (GANITE) Yoon et al.) 2018). counterfactual
regression with Wasserstein distance (CFRwAst ASs) (Shalit et al.
2017). Dragonnet (Shi et al. 2019
and multi-task Gaussian process (CMGP) (Alaa & der Schaar
2017
For the experiments on IHDP and Twins datasets where we have access to both potential outcomes, following (Shalit al. 2017] Yoon et al.) 2018 [Shi et al., 2019 Farajtabar et al. 2020), we use the evaluation metrics- Error in estimation of Average Treatment Effect (EATE) and Precision in Estimation of Heterogeneous Effect (CrEHE). CATE is a global measure in the sense that it measures the error in the estimation of average treatment effect in population. сPEне E a local measure as operates on the error in the estimation of individual treatment effects. For the experiment on the Jobs dataset where we observe only one potential outcome per data point, following (Shalit et al., 2017 Yoon et al. 2018 Shi et al. 2019: Farajtabar et al.) 2020), we use the metric Error in estimation of Average Treatment Effect the Treated (CATT). ). Mathematical definitions and details of these metrics are provided in the Appendix. Following (Shalit et al.) 2017]] Yoon et al. 2018). we report both in-sample and out-of-sample performance w.r.t.
Evaluation Metrics:
Shi et al. 2019
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Under review as a conference paper at ICLR 2023
Datasels
HHD
Twins
Jobs
Metrics +
EATE
EATE
EATT
Methods 4
In-Sample
Out-of-Sample
In-Sample
Out-of-Sample
In-Sample
Out-of-Sample
OLS-1
+ .04
.94 .05
0038 0025
.0069 + .0056
.01 + .00
.08 .04
OLS-2
1441 001
.31 .02
.0399 0025
.0770 + .0059
.01 1: .01
.08 .03
BIR
+
.93 + .05
.0557 + .0036
.0334 + .0092
.01 + .01
.08 +03
k-NN
.14
900 + .05
.0288 = .0011
.0511 + .0039
.21 = .01
.13 + .05
BART
23
.34 .00
.1206 .2366
.1265 .0234
.02 1 .00
.03
Forest
+73
.96 .06
.0499 .034
.0080 .0051
.03 001
.04
Forest
.01
.40 .03
.0866 0035
.0335 + .0083
.03 = .01
.07 .03
BNN
..0
42 1.3
0056 0032
0203 1 .0071
.04 1 01
.09 1 .04
TARNet
2.1
+28
.1188 .0017
.0151 + .0018
.05 -22
.11 .04
MHNET
.14
-37
.108 0008
.0101 + .0002
.04
16:100 .02
GANITE
.43
:49
.0588 0017
.0089 .0075
.01
.6
CFRwAse
1..0
2.
.0112 0016
.284 1 .0032
.04 1.01
.09 1 .03
Dragonnet
166
.29 +31
.0057 + 00033
.0150 + .0003
.04 +00
.04 00
CMGP .11 .10
113
.0124 = 0051
.0443 + .0116
.06 + .06
.09 + .07
NESTER
.04
.09 .07
.0034 .0026
0063 .0033
.06 LL .00
.01
Table 3: Results on IHDP, Twins, and Jobs datasets. Lower better. The numbers are in bold. Second best numbers are underlined. Simple machine learning models, ensemble models, and neural network based models are separated using horizontal lines. See Appendix for further analysis on k-NN results.
VÊPEHE C, EATT in our results. The in ia-sample evaluation is non-trivial since do not observe counterfactual outcomes (all potential outcomes) even during training.
From the results in Table except w.r.t in-sample EATT score in Jobs dataset, NESTER either outperforms or is competitive with the best alternative methods. Our method has the flexibility to learn both complex models that required for small and complex datasets such as IHDP (complex models such as CMGP outperforms simple models such as OLS on IHDP) and to learn simple models solve large and simple datasets such Twins and Jobs (OLS, N-NN often perform better on Twins, Jobs compared to complex models).
Flexibility in Applying Inductive Bias and Program Primitives as Regularizers: Inductive biases, set of assumptions we make to solve ML problem, have significant impact on the ML model performance at test time [Mitchell 1980). For given task, inductive biases are chosen based on the intuition that a particular way of problem-solving is better than others.
These intuitions either come from domain knowledge or from data anal- ysis. As discussed earlier, these inductive biases are implicit in program primitives in the proposed DSL. Using the proposed DSL, for each dataset, NESTER has the flexibility to: (i) Choose or not choose a specific program primitive; (ii) Decide order in which the program primitives are used: and (iii) Use specific program primitive multiple times. This flexibility allows NESTER to use inductive biases differently for different datasets to per- form better. Table shows the best programs synthesized by NESTER for IHDP, Twins, and Jobs datasets. Unlike traditional fixed architectures (e.g.. IPM regularization followed by two head network in CFRNet), NESTER synthesizes path flows (equivalent to different architectures) to solve each dataset. Additional experimental details including analysis on the depth of synthesized programs, impact of the choice of DSL. are provided in the Appendix.
IHDP
if subeet(v. (o..1]) then Corn V alse v..
Twins
Jobs
if subset(v, [o../v then eubeet(y, v..lv[l)
else subset(y, [0../vl]
Table 4: Sample programs learned by NESTER. |v| size of vector v.
CONCLUSIONS
This paper presents a new neurosymbolic programming approach for treatment effect estimation, and also studies why neurosymbolic programming is a good choice for solving such problem. By making an analogy between parameterized program primitives and the basic building blocks of machine learning models in the literature treatment effect estimation,
propose Domain Specific Language on which program synthesis is rooted. Our results and analysis on benchmark datasets with several baselines show the usefulness of the proposed approach. Exploring program primitives corresponding to unexplored heuristics for the treatment effect estimation task is an interesting future direction.

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Under review as a conference paper at ICLR 2023
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12


Reviews:
[{"review_id": "sN1IQ3AD2V1", "reviewer": "Reviewer_Ykau", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper provides a new neurosymbolic treatment effect estimator for estimating treatment effects when the backdoor adjustment formula can be used for identification. Casting the treatment effect estimation problem as a neurosymbolic program is novel, and resulting theory around the ability to approximate continuous functions is interesting. Finally, performance is demonstrated on experimental datasets. ", "summary_of_the_review": "Based on the details explained in the weaknesses section, I do not recommend this paper be accepted. However, I'd be happy to change my mind if the authors can satisfactorily address my questions.", "strength_and_weaknesses": "Strengths: Estimation of Treatment Effects identified via backdoor-adjustment hasn't been explored using neurosymbolic programs before, so there is novelty here. However, please see the discussion of weakness below.\n\nWeakness: When it comes to treatment effect estimation, since we are reasoning with interventions that are not observed and we don't have a ground truth for - it is highly important to be able to provide statistical inference guarantees. Specifically, consistency (I do see a proposition that talks about approximating continuous functions), a favourable rate such as sqrt-n, and valid uncertainty quantification are all desired from a statistical estimator. The theory of influence functions provides us with an Augmented Inverse Probability Weighted (AIPW)estimator which provides sqrt-n rates while allowing for flexible estimation of nuisance parameters using non-parametric estimators. So I don't understand how this proposed neurosymbolic approach fits in with existing approaches for estimating treatment effects. Either this estimator offers lower variance than AIPW, in which case I would like to see a proof. Or if the results of the neurosymbolic estimator are non-asymptotic, I would like to see proofs there as well.\n\nAll in all this paper needs to engage more with the existing literature on treatment effect estimation in order to explain where it improves existing well-established estimators.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "HvWxByNsUQ", "reviewer": "Reviewer_iRvp", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper introduces a neurosymbolic method for dealing the \"treatment effect estimation\" problem. The method is able to synthesize programs in a DSL that was carefully constructed to carry the inductive biases of previous neural approaches for solving the same problem. The programs combine the functionalities of the DSL with neural networks. \n\nThe process in which the programs are synthesized is to use approximations of partial programs (programs with holes) where the holes are replaced by neural networks. \n\nEmpirical results on three different data sets show that the neurosymbolic method is either superior or competitive with other methods in the literature. ", "summary_of_the_review": "Interesting paper, easy to read, and with good results. The method seems to be novel and some of the ideas presented in this paper can possibly be used to solve other problems as neurosymbolic programs are very general. \n\nThe propositions could be false and I am hoping the authors will be able to clarify this issue. ", "strength_and_weaknesses": "Strength\n\nThe use of neurosymbolic programming for solving the treatment effect estimation problem is novel. The way the ideas are presented in the paper allows someone who isn't familiar with the topic to understand enough of the problem to appreciate the results. Overall the paper is very pleasing to read. I thank the authors for it! \n\nI also appreciated the connection between each functionally of the DSL with existing inductive biases from the literature. \n\nFinally, the results are good, perhaps not very strong, but good. The standard deviation in some of the columns are somewhat large and makes one wonder how significant some of the results are, but the results for \"Jobs\" (out-of-sample) are strong with a clear difference between the proposed method and baselines. \n\nWeaknesses\n\nAlthough the paper is very pleasing to read, it is sloppy in the definitions around the propositions of the paper. While I am not absolutely certain (I hope to better understand this issue with the authors' response), I suspect that the propositions do not hold. Here is why. \n\nBoth propositions build on the fact that the heuristic is epsilon-admissible, but the paper never defined (I apologize if I missed this definition) what is the cost of a rule, $s(r)$. This is important because the A* search uses two components: g(n) and h(n) for a given node in the A*'s search tree. The first is the cost traversed from the root of the tree to node n, while the second is the estimated cost-to-go from n to a goal state. The paper defines $\\zeta(P, \\theta)$ for the program $(P, \\theta)$ as the squared loss of what the program produces and the true outcome $y$. \n\nWhile $\\zeta(P, \\theta)$ makes sense in the context of guiding the synthesis (i.e., partial programs with $\\zeta(P, \\theta)$-values smaller will be explored first in search), I don't see how it fits the A* framework because $s(r)$ would have to be defined as something like \"the squared error of each rule used to build the program\". However, I don't see how this is possible as one can't really assign a squared error to a rule; it would make sense to assign an error to a program, but not a rule used to build a program. \n\nThere is a chance that $s(r)$ and $\\zeta(P, \\theta)$ represent two quantities that can't be compared and thus $\\zeta(P, \\theta)$ cannot be an $\\epsilon$-admissible heuristic function for the problem. This would invalidate both propositions of the paper. \n\nMy lack of understanding $s(r)$ is what is holding me from recommending acceptance of the paper. Authors, please help me understand what $s(r)$ is and I will probably be able to connect the points and understand why $\\zeta(P, \\theta)$ is $\\epsilon$-admissible.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "5LDXsARCzdp", "reviewer": "Reviewer_K4Ej", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "This submission presents a methodology for treatment effect estimation by synthesizing causal programs from data. They go on to prove that synthesized programs in the DSL are expressive enough to approximate any continuous function with a small epsilon bound. Finally, they show that the neurosymbolic program synthesis approach outperforms standard effect estimation that assume that the causal graph is fixed.", "summary_of_the_review": "Overall this submission is a promising contribution to the literature. However, it is very important to address comparisons to structure learning based approaches that are prevalent throughout the causal inference literature.", "strength_and_weaknesses": "Strengths:\nOverall the paper is well written and addresses an interesting and important topic in causal inference, effect estimation with unknown structure. Despite my concerns listed below, I see many advantages of the program synthesis approach to effect estimation for reasons that authors discuss. For example, existing methods that combine structure learning with effect estimation lack the interpretability, extensibility, etc. of the synthesis approach presented here. In other words, I am strongly in support of viewing causal models as code, and viewing effect estimation and structure learning as program synthesis.\n\nIn addition to the framing and presentation, the majority of the arguments the authors present are well supported by clear and thorough evidence. The proofs in the appendix appear to be correct and well presented, and the ablation studies are much appreciated.\n\nOpportunities for improvement:\nMy largest concern with this submission, which is a significant one, is that it conceptually and empirically ignores the large literature on causal graph based structure learning. This is a problem for three reasons. \n\n(1) While the authors make standard assumptions in effect estimation explicit, they do not make similarly standard assumptions in causal graph structure learning explicit. For example, do the authors assume faithfulness? See Spirtes et al for more details on these assumptions.\n\n(2) The authors assume no latent confounders, which is sufficient to identify causal effects in the setting where graph structure (or causal program structure) is known apriori. However, non-identifiability is a serious concern when structure is unknown. As a simple example, without strong parametric assumptions (e.g. additive noise) the graphs X -> Y and X <- Y are likelihood equivalent (where here likelihood is a conceptual proxy for the loss used in the paper), but yield dramatically different effect estimates. By choosing a single model, the neurosymbolic approach underestimates the uncertainty in the induced causal effects. It is possible that this might not be an issue with sufficient restrictions on programs (e.g. outcome is always a descendent of treatment), but that must be showed rigorously.\n\n(3) The authors compare empirically only against methods that assume a fixed structure, and where all covariate are pre-treatment. It is important to add two additional baselines. First, it would be useful to show the performance of all of the baseline effect estimation methods when their assumption about covariates being pre-treatment is satisfied. In other words, remove all mediators from their input. Second, it is very important to compare against a handful of baselines that also learn causal structure. One example is to run PC, GES, and/or MMHC (See Spirtes, et al) first, and then to run one of the existing baseline effect estimation methods using the learned structure. Another strong baseline would be to apply the IDA algorithm from Maathuis et al.\n\nSpirtes, Peter, et al. Causation, prediction, and search. MIT press, 2000.\n\nMaathuis, Marloes H., Markus Kalisch, and Peter Bühlmann. \"Estimating high-dimensional intervention effects from observational data.\" The Annals of Statistics 37.6A (2009): 3133-3164.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject