Deep End-to-end Causal Inference
Authors: Tomas Geffner
Javier Antoran
Adam Foster
Wenbo Gong
Chao Ma
Emre Kiciman
Amit Sharma
Angus Lamb
Martin Kukla
Nick Pawlowski
Agrin Hilmkil
Joel Jennings
Meyer Scetbon
Miltiadis Allamanis
Cheng Zhang
Venue: Accepted by TMLR ()

Abstract: Causal inference is essential for data-driven decision-making across domains such as business engagement, medical treatment, and policy making. However, in practice, causal inference suffers from many limitations including unknown causal graphs, missing data problems, and mixed data types. To tackle those challenges, we develop Deep End-to-end Causal Inference (DECI) framework, a flow based non-linear additive noise model combined with variational inference, which can perform both Bayesian causal discovery and inference. Theoretically, we show that DECI unifies many existing structural equation model (SEM) based causal inference techniques and can recover the ground truth mechanism under standard assumptions. Motivated by the challenges in the real world, we further extend DECI to heterogeneous, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Empirically, we conduct extensive experiments (over a thousand) to show the competitive performance of DECI when compared to relevant baselines for both causal discovery and inference with both synthetic and causal machine learning benchmarks across data types and levels of missingness.

Published in Transactions on Machine Learning Research (06/2024)
Deep End-to-end Causal Inference
Tomas Geffner
Javier Antoran 2
Adam 3oster Wenbo Gong² Chao Ma
Emre K
Amit Sharma Angus Lamb 4 Martin Kukla²
Nick Pawlowski
Agrin Hilmkil Joel Jennings
Meyer Scetbon
Miltiadis Allamanis
Cheng Zhang
University of Massachusetts Amherst
University of Cambridge
Microsoft Rescarch
G-Rescarch
Wembooong@microooff..cm
Reviewed on OpenReview: http::///ppaaanrrss...ee///////p
Abstract
Causal inference is essential for data-driven decision-making across domains such as business engagement, medical treatment, and policy making. However, in practice, causal inference suffers from many limitations including unknown causal graphs, missing data problems, and mixed data types. To tackle those challenges, we develop Deep End-to-end Causal Inference (DECI) framework, flow-based non-linear additive noise model combined with variational inference, which can perform both Bayesian causal discovery and inference. Theoretically, we show that DECI unifies many existing structural equation model (SEM) based causal infer- ence tochniques and can recover the ground truth mechanism under standard assumptions. Motivated by the challenges in the real world, we further extend DECI to heterogeneous, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Empirically, competitive performance of DECI when compared to relevant baselines for both causal dis- covery and inference with both synthetic and causal machine learning benchmarks across data types and levels missingness.
conduct extensive experiments (over thousand) to show the
1 Introduction
Causal-aware decision making is pivotal in many fields such as economics (Zhang & Chan 2006 Battocchi ct al. 2021) and healthcare (Bica et al. 2019 Huang. 2021 Tu et al.] 2019). For example, in healthcare, caregivers may wish to understand the effectiveness of different treatments given only historical data. They aspire to estimate treatment effects from the observational data, which we refer to as causal inference task. Unfortunately, there exist several challenges that prevent standard causal inference techniques to be applied in the above real-world scenario.
First, existing causal inference methods for estimating causal quantitics from data commonly assume plete a priori knowledge of the causal relations (i.e. causal graph), which is rarely known, especially with many variables. Secondly, there may exist missing values and mixed data types in the history dataset, which poses another layer of difficulty to apply the causal inference methods.
To lackle the above challenges without latent confounders, we propose a Deep End-to-end Causal Infer- ence (DECT) framework[] a flow-based non-linear additive noise model combined with variational inference, where end-to-end represents the capability of performing both Bayesian causal discovery and inference under missing values and mixed-type data. Figure illustrates the pipeline of DECI framework.
To enable this, we specifically contribute the following:
"Equal contribution. 1 Contributed during interuship residency in Microsoft Research. Code for reproducing experiments available httpp://ssttua.cccoccttttt/ttt/s Latest version of DFCT is s/tttps://github httpp////gggthuu..ooo/ttttotttos com.://ccccrroootf//cussscss

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X
22
X3
X4
X
X1
22
X3
X4
0.1
1.2
0.5
0.9
0.9
0.2
1.5
0.3
1.1
1.0
X
X
X1
(1) Observe data correspond-
2) Learn the causal relation-
(3) Learn the functional rela- tionships among variables.
ing to D variables.
ships among all variables.
X1
X2
X3
X₂
E[Xs/do(XX X2
X
X
do(X2 = 4.94)
(4) Select intervention and target variables.
(5) Estimate causal quantities such as ATE and CATE.
(6) Make optimal decisions and take actions.
An overview of the Deep End-to-end Causal Inference pipeline compared to traditional causal discovery and causal inference. The dashed line boxes show the inputs and the solid line boxes show the outputs. In causal discovery, & user provides observational data (1) as imput. The output is the causal relationship (2) which are either DAGs or partial DAGs. In cansal inference, the user needs to provide both the data (1) and the causal graph (2) as input and provide a causal question by specifying treatment and effect (4). Then, model is learned and outputs the causal quantities (5), which helps decision making (6). In this work, the proposed DECI can perform both causal discovery and inference, which only requires the user to provide the observational data (1) and specify any causal questions (4), and output both the discovered causal relationship (2) and the causal quantities (5) that help in decision making (6).
Figure 1:
DECI formulation for discovery and inference. We propose an end-to-end framework that combines non-linear additive noise SEM with variational inference to infer the graph posterior and learn functional parameters. Further, propose flexible noise based on spline transformation (Durkan et al. 2019). With the learned SEM, we derive how to use it for Aimulation-basee treatment effect estimation (i.e. (conditional) average treatment effect ((C))TE))
extend the noise distribution beyond standard Gaussian and
2. Theoretical analysis of DECI. To show the theoretical advantages of DECI, first prove that under correct model specification and certain assumptions, DECT asymptotically recovers the true data- generating mechanism and correctly estimates the treatment effect. Additionally, we show DECI is flexible model that unifies many existing SEM based causal methods, such as Nolears (Zheng
et al.
2018a 2020), Grandag [Lachapelle et al. 2019), MissDAG (Gao et al. 2022 and others (g
al.
2019 2020
3. Extension of DECI to real-world scenario. Due to the unifying framework, we can equip DECI to perform missing value imputation similar to MissDAG. addition, we also show how to handle mixed-type data (continuous and categorical).
4. Extensive empirical evaluations. To verify the effectiveness of DECI, we conduct extensive exper- iments (including synthetic and real-world benchmarks), along with a range of existing discovery and inference algorithms. DECI consistently outperforms the baselines in the discovery tasks and performs very competitively in causal discovery + inference tasks.
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2 Related Work and Preliminaries
Related Work. Our work is related to both causal discovery and causal inference research. Approaches for causal discovery from observational data can be classified into three groups: constraint-based, score- based, and functional causal models (Glymour et al.) 2019). Recently, Zheng et al. (2018a) framed the directed acyelic graph (DAG) structure learning problem as a continuous optimization task. Extensions [Lachapelle et al. 2019 Zheng et al. 2020 Gao et al., 2022) employ nonlinear function approximators, like neural networks, to model the relationships among connected variables. Our work combines this class of approaches with standard causal assumptions (Ng et al., 2019), to obtain our main theorem about causal graph learning. We extend functional methods to handle mixed data types and missing values. In particular, MissDAG proposed to use the expectation maximisation framework to perform missing value imputation and causal discovery at the same time. It represents a special case of DECI with Gaussian noise and point estimate graph. Therefore, we also propose the similar technique for imputing missing values. Outside of functional causal discovery, functional relationships between variables (see Figure 33) are typically not learned by discovery algorithms (Spirtes & Glymour 1991). Thus, distinct models, with potentially incompatible assumptions or inputs, must be relied upon for causal inference. However, when DAG cannot be fully identified given the available data, constraint and score- boore-baed methods often return partially directed acyelic graphs (PDAGs) or completed partally directed acyclic graphs (CPDAGs) (Chickering 2002 Andersson et al. 1997). Another line of works consists in estimating posterior distribution over the DAGs from observed data. The central challenge in this Bayesian setting (Friedman & Koller 2003 Heckerman et al. 2006; Tong & Koller 2001) lies in the support of the posterior which is a union of exponentially growing (discrete) DAGs. Prior works have used Markov Chain Monte Carlo (MCMC) to directly sample DAGs or
bootstrap traditional discovery methods [Chickering] 2002 Murphy 2001 Tong & Koller 2001), but these methods are typically limited to linear models which admit closed- marginalization over continuous parameters. Recent advances have begun to utilize gradient information for more efficient inference <Lorch) ct al. 2021 Gong et al. 2021). ). In this work, we leverage these recent advances in DECI to estimate a
distribution over DAGs.
Causal inference methods assume that either the graph structure is provided (Pearl 2009b) or relevant structural assumptions are provided without the graph (Imbens & Rubin 2015). Causal inference can be decomposed into two steps: identification and estimation. Identification focuses on converting the causal estimand (e.g. P(Y]do(X = =z), W)) into estimand that can be estimated using the observed data distribution (e.g. P(Y|X, W)). Common examples of identification methods include the back-door and front-door criteria (Pearl 2009b), and instrumental variables (Angrist et al.) 1996). Causal estimation computes the identified estimand using statistical methods, such as simple conditioning, inverse propensity weighting [Li ct al., 2018), or matching (Rosenbaum & Rubin 1983; Stuart 2010). Machine learning- basod estimators for CATE have also been proposed (Chernozhukov el al. 2018 Wager & Atheey] 2018). Recent efforts to weaken structural assumption requirements (Guo & Perkovic 2021] Jung et al.) 2021] allow to perform causal inference for PDAGs and CPDAGs. However, both require prcrequisite graphs input (i.e. partially ancestral graph maximally oriented PDAG), whereas DECI can infer distribution over DAGs from observational data. In addition, DECI, once learned from data with satisfied assumptions, can provide identification on all causal quantities, Whi & Perkovic (2021 Juun et al. (2021) either assumes identifiability at the beginning or provides a set identification on total causal effect. (Jung et al.) (2021) also leverage double machine learning (Chernozhukov et al., 2018), a linear model, as the estimation for causal quantitics, compared to the nonlincar neural network of DECI. PPawlosski et al. (2020) proposed a variational approach, DSCM, to infer the posterior noise distribution during counterfactual computation. However, they assume causal graph as priori and extend beyond the additive noise structure. Our method is orthogonal to it and can infer both cansal graph and counterfactuals.
Structural Equation Models (SEM). Let x = (#1,..., ID) be a collection of random variables. SEMs (Pearl 2009b) model causal relationships between the individual variables z, Given a DAG G on nodes (1...., D), x can be described by x; F, (xp((xc)): 2.), where 2 is an exogenous noise variable that is independent of all other variables in the model, pa(i: G) is the set of parents of node in G, and F, specifies how variable 2; depends its parents and the noise 2, In this paper, we focus on additive noise SEMs,
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also referred to additive noise models (ANM), i.c.
Fs (Xpa(t:))) 24) fo (X(m()t)) + == + in vector form.
(1)
Here, omit the dependency of function Ji to its parameters 0. To represent the graph G, use binary adjacency matrix. In this paper, we interchangeably use G to represent graph and adjacency matrix, where Giy = represents i J.
Average Treatment Effects. The ATE and CATE quantities allow us to estimate the impact of our actions (treatments) (Pearl 2009b). Assume that xr (with C {l...., ,.,,, are the treatment variables; the interventional distribution is denoted p(x do(x; a)). The ATE and CATE on targets Xy for treatment x given reference xy=b, and conditional on x for CATE, are given by
ATE(a, b) (d)(x))) [xy] Epxx [do(x)=b)) XY], and CATE(a, blc) Ep(xy xx)(xx=a)xxx=0))) [xy] Ep(xy b))
(2)
We consider the common scenario where the conditioning variables are not caused by the treatment. Sepcifi- cally, we focus the scenario where the treatment Xy is not the cause of the conditioning set xc, i... there is no directed path from T to C in G.
3 DECI: Deep End-to-end Causal Inference
We introduce DECI, an end-to-end deep learning- learning-based causal inference framework. DECI learns distribu- tion over causal graphs from observational data and (subsequently) estimates causal quantities. Section 3.1 describes our autoregressive flow based ANM SEM. Section 3.2 lays out the conditions under which DECI will recover the true causal graph given enough observational data (Theorem []] Section 3.3 shows how the gencrative model learnt by DECI can be usod simulate samples from intervened distributions, allowing for treatment effect estimation. Section 3.4 extends DECI's real-world applicability by adding support for non- Gaussian exogenous noise, mixed type data (continuous and discrete), and imputation for partially observed data.
3.1 DECI and Causal Discovery
DECI takes a Bayesian approach to causal discovery (Heckerman al. 1999). We model the causal graph G jointly with the observations x' x'....,x"' x.
as
(x ,...,,'''' N = pG) II Po)
(4)
We aim to fit 0, the parameters of our non-linear ANM, using observational data. Once this model fitted, the posterior "G.,'"...,xx"" characterizes our beliefs about the causal structure.
Prior over Graphs. The graph prior p(G) should characterize the graph as DAG. We implement the prior as
P(G) x exp where we weight the DAG penalty by a and 0s Ile indicates the Frobenius norm and M(G) = CoG)
(5)
with O the Hadamard product. The squared term is from the augmented Lagrangian @Neiirvvsky 1999). The advantage is that it closely approximates the solution of the constraint problem with the solution of uncoustrained problem without increasing P to infinity, P, are gradually increased during training following augmented Lagrangian scheme, ensuring only DAGs remain at convergence. We introduce prior knowledge about graph sparseness by penalising the norm A,I with X₂ a scalar. We can also incorporate domain knowledge to specify a more informative prior, see appendix E.6 for full details.
Likelihood of Structural Equation Model. Following Khemakhem al. (202)), we factorize the ob- servational likelihood pe(x"IG) in an autoregressive manner. Rearranging the ANM assumption eq. ( we
have (x(x;)) 0) fa(x; 0). The components of independent. If we have distribution Pz; for
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component 2, where 2 is the exogenous noise for variable Is, then can write the observational likelihood 88
D po(x*| (G) Pz (gc ;0)) II Ps. (oc,((x";0))
(6)
where the first equality is from the change of variable formula since 9G is the invertible transformation between x, z and we omitted the Jacobian-determinnnt term because it is always equal to one for DAGs G (Jemma 3 in appendix C 9G, represents the jth elements of its output. The choice of fo R2 - Rd must satisfy the adjacency relations specified by the graph G. If there is no edge j + i in G, then the function [((x) the i-th component of the putput of fa(x) must satisfy 0f (x)(x)/0r = 0. We propose a flexible parameterization that satisfies this by setting
d f(x) S Si Σ Gz EgNB(oa)) ((1)))
(7)
where Gj,a E {0, 1) indicates the presence of the edge + i, and l, and Sa (i= (i= I,..., d) are MLPs. A naïve implementation would require training 2D MLPs. Instead, we construct these MLPs so that their weights are shared across nodes as G(.)) = (( and (s(-) = ((u;,)), with u, E RR a trainable embedding that identifies the source and target nodes respectively.
Exogenous Noise Model Pz- We consider two possible models for the distribution of z: 1) simple Gaussian (-) N (-10, of), where per-variable variances of are learnt. 2) A flow (Rezende & Mohamed 2015
On, (zx)
(z) = (x; (z)1000)
(8)
02,
We choose the learnable bijections K; to be a rational quadratic spline Durkan et al. independently across dimensions. We do not couple across dimensions since our SEM requires independent noise variables. Spline flows are significantly more flexible than the Gaussian distributions employed in
2019
parametrised
previous work (Lachapolle et al.) 2019 Ng et al. 2019 2020; Zheng et al.) 2018a 2020). Here, we use Gaussian distribution as the base, but one can easily replace it with other forms of distributions, as long as there exists differentiable density functions.
Optimization and Inference Details. The described model presents two challenges. First, the truc posterior over G is intractable. Second, maxinun-likelllood cannot be used to fit the model parameters, due to the presence of the latent variable G. We simultaneously overcome both of these challenges using variational inference (Blei et al. 2017 Jordan et al. 1999 Zhang et al. 2018). We define a variational distribution (G) to approximate the intractable posterior po(G ,...,"") xx) and use it to build the ELBO, given by
ELBO(0, o) = (G)) log Po (G)
H(4)) S log po x )M
(9)
(G) takes the form of eq. (6) (derivation in Appendix B.3 We model (G) the product of independent Bernoulli distributions, one for each potential directed edge in G. We use the ENCO parameterization (Lippe et al.) 2021) to represent the probability of each potential directed edge as g0 (G) II., Ber(Ga) ((T))) where a(-) is the sigmoid function, 7 E DxD represents the existence of an edge, and DxD with dij --0; describes the edge orientation. (-) indicates the sigmoid function. Therefore, the two orientation probabilities will always sum to one. A high probability direction indicates low probability of the other direction. This can help to prevent the length-1 cycles. In addition, the existence probability is casier to learn compared to the orientation since it does not need to consider dagness. Thus, the separation of existence and orientation probability allows for better control of gradient for easier optimization. The SEM parameters / and variational parameters Ø are trained by maximizing the ELBO. The Gumbel-softmax trick (Jang et al., 2016 Maddison et al.) 2016) is used to stochastically estimate the gradients with respect to 6. Appendix B.1 details the full optimisation procedure.
where (gg)) represents the entropy of the distribution Is and (xx"
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Unified View of Functional Causal Discovery. We note that, like DECI, many functional causal discovery methods (Lachapelle et al.) 2019 Ng et al. 2019 2020 Zheng et al. 20 2020 can be seen from probabilistic perspective as fitting an autoregressive flow (with a hard acyelicity constraint) for different choices for the exogenous noise distribution Pz and transformation function f. We expand on the details of this perspective and formalise in Appendix C In a nutshell, we show that the training objective and model parametrization of many existing causal discovery methods can be viewed as a special case of DECT's parametrization and objective function. In particular,
Notears (Zheng et al., 2018a) uses standard isotropic Gaussian for Pa and linear transformation for [(x 0).
Notears-MLP (Zhnng et al., 2020 uses standard isotropic Gaussian for Pa and D independent multi- layer perceptrons (MLP), one for each (x).
Notears-Sob (Zheng et al., 2020) uses a standard isotropic Gaussian for pz and weighted linear com- bination of nonlincar basis functions for cach fx))..
GAE (Ng et al., 2019) uses standard isotropic Gaussian for Pa and GNN for fi(x).
Grandag (Lachapelle et al., 2019) uses factorized Gaussian with mean zero and learnable scales for Pz and D MLP, one for each fx)..
Golem Ng et al., 2020). - This is a linear method whose original formulation was already in a probabilistic perspective, using linear transformation for fx(x).
Therefore, we can see DECI framework provided a unified view on the above methods. DECI also extends the above by allowing the f and noise distribution to be MLP and potentially non- non-Gaussian, respectively, which makes it the most flexible member of this family.
3.2 Theoretical Considerations
One key statistical guarantee of DECI is to show that optimizing the variational objective eq. 9 leads to the recovery of the ground truth data generation mechanism under the infinite data limit.
We formalize this guarantee in Theorem which states that under certain assumptions, eq- is indeed a valid objective function. Specifically, Theorem ! requires the following assumptions:
Non-trinial function, the function f is non-invertible, 3rd differentiable and is not constant w.r.t. any of its inputs. This assumption can be met by using MLPs.
Regularity of noise distribution, the noise distribution has differentiable probability density function with bounded likelihood.
Correct specification, there exists parameter 0*, such that po. po- (æ; Go) is the gentratnnn distribution, where G is the ground truth causal graph. Note that although 0" may not be equal to the ground truth parameter, we do not care about the identifiability in parameter spaace, but in functional space. Namely, the equality of functions.
No latent confounder, this is also called causal sufficiency where there are no un-observed confounders in the data.
The above assumptions in fact imply some of the common causal assumptions, e.g. Markov property, causal minimality, and exclude some abnormal cases (e.g. linear Gaussian SEMs) for model structural identifiability. Refer to Appendix A for formal statements and proof.
Theorem (DECI recovers the true data generating process). Assume DECT's underlying SEM belongs to
continuous additive noise structural equation models with (1) non-invertible, 3rd-order differentiable function between variables that. are not conslant. w.r.l. any of their inpuls (Le. non-trivial); (2) differentiable noise density and bounded log- likeekhoodd then, under correctly specified model, infinite data limit and no latent confounders in data generating mechanism, the solution (6' dy())) from marimizing the ELBO (eg. (9)
satisfies d' (G) 5(G = G') where G' is a unique graph. In particular, G' =o and po (x; G') p(x: Go)
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8 represents the Dirac delta function. Note that we do not claim the identifiability of 0 in the parameter space, but rather on the functional space. To demonstrate the trend of graph posterior converging to a single graph with increasing dataset size, we plot its entropy with different dataset sizes on Figure G in appendix A.5
Note that Theorem 1 only proves that DECT can recover the true graph and data generating distributions when ELBÒ is maximized under infinite data regime. We do not prove the tendeney on whether the optimization can maximize the ELBO, which depends on properties of the chosen optimization and is out of the scope of this work. One potential future direction to strengthen Theorem i to prove the convergence rate w.r.t the number of data points, similar to the variational Bernstein Von-Mises theorem (Wang & Blei 2018). Another point worth noting is that Theorem | does not require the uniqueness of the model parameters 0, as we only prove the recovery in the distributional sense. The claim is still valid even if we have a set of parameters, as long as they induce the same distribution as the ground truth distribution under the true graph.
Model Misspecification One of the key assumption for theorem is the model specification. When the model is misspecified, theorem no longer holds. This means that there is no guarantee the recovery of the true generation mechanism and graph. There are some existing theoretical works on variational inference under the misspecified model (Wang & Blei 2019: Knoblauch 2019). Wang & Blei (2019) claims that even with model misspecification, the inferred postcrior will collapse to a point that minimizes the KL divergence between the resulting and ground-truth generation mechanism. However, their methodology cannot be directly applied to our setting, since they assume that the random variables are continuous rather than discrete like graphs. This opens new rescarch opportunity to prove similar guarantee for graphs well. Despite that, we can still leverage the universal approximation capacity of MLPs (Hornik et al.) 1989 meaning that they can approximate continuous functions arbitrarily well. This flexibility gives us higher chance that this assumption indeed holds.
3.3 Estimating Causal Quantities
We now show how the generative model learned by DECI can be used to evaluate expectations under interventional distribu- tions, and thus estimate ATE and CATE. As explained above, DECI returns (G), an approximation of the posterior over graphs given observational data. Then, interventional distribu- tions and treatment effects can be obtained by marginalizing over graphs
0.25
Asprox maded
Arproc mnted ATE (Geaph
0.20
True
Apprux mated ATE [Graph 2)
Estimateed
0.15
E (C) (x(x))do(xr=a) G)], (ATEa,) and E (g,,cICCATT(E blc, G)].
0.35
(10)
This can be seen as probabilistic relaxation of traditional causal quantity estimators. When one has observed enough data to be certain about the causal graph, i.e, (G) = 8(G- Figure 2:
Marginalizing ATE estimates from graph and exogenous noise samples.
G.), our procedure matches traditional causal inference. We go on to discuss how DECI estimates AC)
Estimating ATE. After training, we use the model learnt by DECI to simulate new samples from (x)). We sample graph G ~ (G), and set of exogenous noise variables z ~ Pz. We then input this noise into the learnt DECI structural equation model to simulate x, by applying eq. [ and eq. 7 on z in the topological order defined by G. However, ATE estimation requires samples from the interventional distribution p (xxr do(xT = b), G). These can be obtained by noting that
p
= b), G) = (x = Gdo(er)),
where Gao(xr) is the "mutilated" graph obtained by removing incoming edges to xy. Thus, samples from this distribution can be obtained by following the sampling procedure explained above, but fixing the values
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Xy = and using Gao(xe)) instead of G. Finally, we use these samples to obtain Monte Carlo estimate of the expectations required for ATE computation eq- 2 When the posterior go(G) has not collapsed to one graph, these samples are from a mixture distribution, translating the uncertainty in the discovered graph into uncertainty in ATE estimates. This is illustrated in Figure 2
Estimating CATE. Under the assumption that xr is not a cause of xc, we can estimate CATE by sampling from the interventional distribution p (xxr)do(x)). G) and then estimating the conditional distribution of Xy given x To make this precise, we let Y = X (TUC) denote all variables that we do not intervene condition on. Conditional densities
pa (x b/ Gdo(xr))
(x d)(xx=b), xc=c, G) px(xc=c|xy=b, Gdo(xr))
(1))
are not directly tractable in the DECI model due to the intractability of the marginal pa p=(xo=c\xy=b, Ga However, we can always sample from the joint interventional distribution Pa xc/x,, Gdo(xr))- We use samples from this joint to train a surrogate regression model he to the relationship between Xe and xy. Specifically, we minimize the square loss
E Epo(Xy xe xe
[|lxy - h=()))
making ha approximate the conditional mean of xy. We choose hG to be basis-function linear model with random Fourier basis functions (Yu et al., 2016 As illustrated, in Figure 3 we train two separate
surrogate models, one for our intervention XT and one for the reference XT = b. We estimate CATE
as the difference between their outputs evaluated at Xc = This process is repeated for multiple posterior graph samples G Jø allowing us to marginalise the posterior graphs
Eyg( [a (Xc =c) bny (xc c))
(12)
Theoretical considerations. We previously showed that variational inference asymptotically recovers the correct SEM parameters 0* in Theorem | We further show that a correctly specified DECI model satisfies causal effect identifiability and positivity, mcaning that treatment effects will be correctly estimated in the large data limit (refer to Appendix A.4).
General ECI Framework. The probabilistic treatment of the DAG, and the re-use of functional causal discovery generative models for simulation-based causal inference are principles that can be applicd beyond DECI. Constraint-based (Spirtes Glymour 1991) and score-based (Chickering) 2002) discovery methods often output set of DAGs compatible with the data, i.e., PDAG or CPDAG. It is natural to interpret these equivalence classes as uniform distributions over members of sets of graphs. We can then use eq. (10) to estimate causal quantitics by marginalizing over these distributions. The quantities inside the expectations over graphs can be estimated using any existing causal inference method, such as linear regression (Sharma et al. 2021), Double ML (Chernozhukov et al. 2018), etc. Our experiments explore combinations of discovery methods that return graph equivalence classes with standard causal inference methods. We take expectations over causal graphs since these return the quantity that minimises the posterior expected squared error in our ATE estimates while noting that the best statistic will be application dependent.
3.4 DECI for Real-world Heterogeneous Data
We extend DECI to handle mixed-type (continuous and discrete) data, and data with missing values, which often arise in real-world applications.
Handling Mixed-type Data. For discrete-valued variables, we remove the additive noise structure and directly parameterise parent-conditional class probabilities
discrete Pg
(zi(xpa(t:3)) G) = P. (xmatiay;)) (z:),
(13)
2Subscript G allows differentiating surrogate models on samples from different, graphs drawn from 9g-
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Obeervation distributionn p(zy, .R)) Intervened distribution play. zcldo(er))
DECI Learnt Graphas
Interverrion Reference
ay.
True CATE:
-2-
-
The ground truth distribation
P(G = 0.6)
Reftrence
(Correct Graph)
Go ay
Go CATE: 2.32
dan
P(G. = 0.6)
DECI CATE: 1.84
Intevention
Reforence
P[G 0.4)
G1 aty
Gj CATE: 1.12
P[G] 0.4)
20
DECI estimation
Figure 3: DECT CATE estimation on the CSuite Symprod Simpson dataset. Left: The DECI graph posterior has two modes with p(G) = 0.6 for the correct graph and p(G) = 0.4 for an alternative possibility with some incorrect edges. Middle: we display the joint distribution of conditioning and effect variables in the observational setting and under interventions
Xy. DECI captures the observational density well. Right: interventional distributions with their conditional means xc marked with crosses. DECI predicts conditional expectations by fitting functions from xe to xy and evaluating them at DECI outputs CATE by marginalizing the result over possible graphs.
where Ps (Xpa(to): 0) is normalised probability mass vector over the number of classes of Ty obtained by applying the softmax operator to fi(Xpa(t))) This means that for discrete variables, the output of Si is a vector of length equal to the number of classes for variable i. This approach gives a valid likelihood for Go()"(G) which we use to train DECI. However, since the full generative model is no longer an ANM, we cannot guarantee that Theorem appliess this setting.
Handling Missing Data. We propose an extension of DECI to partially observed data." We use x to de- note the observed components of x", x", to denote the unobserved components, and their joint density in the observational environment is pa (xx; x!)) We approximate the p(G, x"xx") with the variational distribution,
9 (G, x ...,..... x x |x!................. x (G) II qo (x))
which yields the following learning objective
Il pxxx x+10]
ELBO(8,0, = H(go) + Σ ((qx(x(xx²)) Egg..
log pap
(14)
We parameterize the Gaussian imputation distribution ge. x using an amortization network (Kingma & Welling 2013), whose input is x and outputs the mean and variance of qu (x?()))
4 Experiments
We evaluate DECI on both causal discovery and causal inference tasks. A full list of results and details of the experimental setup in Appendices and El
We assume that values missing (completely) at random, common setting Ma
2018 Rubin
1976
Stekhoven & Bühlmann 2012 Strobl al. 2018).

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Published in Transactions on Machine Learning Research (06/2024)
4.1 Causal Discovery Evaluation
Datasets.
We consider synthetic, pseudo-real, and real data. For the synthetic data, we follow Lachapelle et al. (2019) and Zheng et al. 20200] by sampling DAG from two different. random graph models, Erdös-Rényi (ER) and scale-free (SF), and simulating each ANM I; f((p(a(t))) ) + 2i, where fi is nonlinear function (randomly sampled spline). We consider two noise distributions for 24. standard Gaussian and a more complex one obtained by transforming samples from a standard Gaussian with an MLP with random weights. We consider the number of nodes d E {16, 64) with the number of edges e E (d.. The resulting datasets are identified as ER(d, e) and SF e). All datasets have n=5000 training samples.
For the pseudo-real data we consider the SynTReN generator (Van den Bulcke et al. 2006), which cre- ates synthetic transcriptional regulatory networks and produces simulated gene expression data that mimics experimental data. We use the datasets generated according to Lachapelle et al. (2019) (d-20), and take n=400 for training. Finally, for the real dataset, we use the observational protein measurements in hu- man cells from Sachs et al.] (2005), in which we sampled a training set with n=800 samples and d=11.
Baselines. We run DECI using two models for exoge- nous noise: a Gaussian with learnable variance (identi- fied DECI-G) and a spline flow (DECI-S). We com- pare against PC (Kalisch & Bühlman 2007), (linear) Notears Zheng et al., 2018a), the nonlinear variants Notears-MLP and Notears-Sob (Zheng et al.) 2020 Grondag Lachapelle et al.) 2019), and ICALINGAM Shimizu et al. 2006). When a CPDAG the out- put, e.g., from PC, we treat all possible DAGs un-
PC
DACLAPO
Noneer Sep
10
der the CPDAG as having the same probability. All
baselines are implemented with the gcastle package
S. E U.C. 8
Zhang ct al. 2021
Causality Metrics. We report F1 scores for adja- cency, orientation (Glymour et al. 2019 Tu et al. 2019 and causal accuracy (Claasson Heskes 2012). For DECI, we report the expected values of these metrics estimated over the graph posterior.
EPEI8 65
BPvel m4
ERBA df
DRII M
BFIM 66
Spttfee
Pooto
Figure 4: Causal discovery on benchmark datasets. The label (PO) corresponds to running DECI with 30% of the training data missing. For readability, the DECI results by are connected with with soft lines. The figure shows mean results across five different random seeds.
Figure shows the results for the data generated with non- Gon-Gaussian noise. We observe that DECI achieves the best results across all metrics. Additionally, us- ing the flexible spline model for the exogenous noise (DECI-S) yields better results than the Gaussian model (DECT-G). This is expected, as the noise used to generate the data is non-Gaussian. For Gaussian noise (see Figure both DECI-S and DECI-G perform similarly. Moreover, when data are partially observed (PO), the strong performance of DECI remains, showing that DECI can handle missing data elficiently.
4.2 End-to-end Causal Inference
We evaluate the end-to-end pipeline, taking in observational data and returning (C)ATE estimates.
Datasets. We generate ground-truth treatment effects to compare against for the ER and SF synthetic graphs that were described in Section 4.1 applying random interventions on these synthetic SEMs, ensuring at most 3 edges between the intervention and effect variables. For more detailed analysis, we hand-craft a suite of synthetic SEMs, which we name CSuito. CSuite datascts clucidate particular features of the model, such as identifiability of the causal graph, correct specification of the SEM, exogenous noise distributions, and size of the optimal adjustment, set.. We draw conditional samples from CSuite SEMs with Hamiltonian
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Twins ATE
IHDP ATE
Twins CATE
IHDP CATE
DGa
DGa+L
DGa+N
DSp+L
DSp+N
PC+L
PC+N
0.02
0.04
0.06
0.08 0.075
0.100
0.125 0.150 0.175
0.00
0.02
0.04
0.06
0.08 0.16
0.18
0.20
0.22
0.24
ATE RMSE
ATE RMSE
CATE RMSE
CATE RMSE
Figure 5: Box plots showing end-to-end ATE and CATE estimation error on the semi-synthetic Twins and IHDP datasets with different method combinations. Method acronyms are as in Table Box plots show the median and interquartile range over 5 seeds.
Monte Carlo (HMC) (Neal ct al. 2011); allowing us to evaluate CATE. Finally, we include two semi-synthetic causal inference benchmark datasets for ATE evaluation: Twins (twin birth dalasets in the US) (Almond et al., 2005) and IHDP (Infant Health and Development Program data) all experimental details.
Hill 2011 See Appendix D for
Baselines.
Table 1: Method rank on different (CSuite, Twins, IHDP and ER/SF) datasets, ranking by median ATE RMSE. We present +1 s.e. of the rank over 27 datasets. Supporting data in Table 3 Bold indicates the possible top methods, accounting for error bars. We treat methods with access to the true graph separately.
To thoroughly evaluate end-to-end infer- ence, we consider different ways of combin- ing discovery and inference algorithms. For use trained model to im- mediately estimate (C) We also con- sider using the learned DECI graph pos- terior in combination with existing meth- ods for causal inference on a known graph: DoWhy-Linear (T) and DoWhy-Nonlinear (N) (Sharma al. 2021) which imple- ment linear adjustment and Double Machine Learning (DML) (Chernozhukov el al. 2018) methods for backdoor adjustment respec- tively. We also pair other discovery meth- ods with DECI and DoWhy treatment effect estimation, namely the PC algorithm as a baseline and the ground truth graph (when available) as check. We evaluate end-to- end causal inference on all valid combina- tions that arise from combining discovery methods in (DECI-Gaussian (DGa), DECI- Spline (DSp), PC, and True graph (T)} with causal inference methods in (DECI-Ganssin (DGa), DECI- Spline (DSp), DoWhy-Lineal (L), DoWhy-Nonling (N)).
DECI,
Method
Mean rank
DECI Gaussian (DGa)
6.26 = 0.60
DECI Gaussian DoWhy Linear (DG++L)
8.37 = 0.50
DECI Gaussian DoWhy Noulinear (DGa+N)
8.52 = 0.51
DECI Spline (DSp)
6.04 = 0.68
DECI Spline DoWhy Linear (DSp+L)
7.78 = 0.60
DECI Spline DoWhy Nonlinear (DSp+NN
6.63 - 0.66
PC + DoWhy Linear (PC+L)
8.87 = 0.41
PC + DoWhy Nonlinear (PC+N)
7.54 = 0.45
True graph DECI Gaussian (T+DGa
3.74 = 0.47
True graph DEC1 Spline (T+DSpp
4.19 = 0.56
True graph DoWhy Linear (T+L)
1.87 = 0.58
True graph DoWhy Noulinear (T+N)
5.20 = 0.71
Metrics. We report RMSE between (C)ATE estimates and the ground truth.
Table provides high-level summary of our results. For each dataset, we estimated the ATE using each combination of methods, computed the RMSE and took the median over random seeds. We then ranked methods for each dataset (with 1 being the best) and aggregated over the 27 datasets. We find that DECI Spline has the overall best, (lowest) rank. RMSE scores are in Appendix
In Table 2 we present detailed results for six CSuite datasets. Lin. Exp two node lin- ear SEM with exponential noise, only DECI Spline can recover the true graph, ATE estimation qual- ity is similar for different estimator once the true graph is found. Nonlin. Gauss is two node non-linear SEM with Gaussian noise, only DECI can fit the highly non-linear functional re- lationship, with equal performance between DECI-Gaussian and -Spline. Large backdoor is a larger nonlinear SEM with non-Gaussian noise in which adjusting for all confounders is valid, but of high variance. For DECI-Spline, which performs well on discovery, the ATE cstimation is bost using DECI, as DoWhy takes the maximal adjustment. set thereby increasing estimator variance.
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Weak arrows is similar SEM to Large backdoor, except that maximal adjustment set is now necessary. Here, DECI-Spline is best for discovery, but is somewhat less ac- curale for ATE estimation given the right graph. Nonlin. Simpson is an adversari- ally constructed dataset where 1) the true graph is theoretically identifiable, but dif- ficult to discover in practice, 2) ATE esti- mation is very poor given the wrong graph (Simpson's paradox). All methods perform equally badly. Symprod Simpson a sim- ilar but slightly casier dataset, for which DECI-Spline with DML does well.
Table 2: Median ATE RMSEs from 20 seeds for six CSuite
datasots. Acronyms as in Table G
Nonlin. Gauss
Large backdoor
Nonlin. Symprod arrows Simpson Simpson
Weak
Lin. Exp
DGa
1.029
0.042
0.213
1.097
1.995
0.318
DGa+L
1.031
1.522
0.141
1.108
1.994
0.695
DGa+N
1.031
1.532
0.331
1.108
1.994
0.487
DSp
0.022
0.043
0.031
0.189
1.997
0.427
DSp+L
0.001
1.522
0.091
0.110
1.994
0.819
DSp+N
0.002
1.532
0.232
0.064
1.994
0.160
PC+L PC+N
0.516
1.532
1.690
1.108
1.994
0.487
0.517
1.532
1.690
1.108
1.994
0.487
T+DGa
0.073
0.034
0.167
0,255
0,404
0,101
T+DSp
0.028
0.034
0.035
0.128
0.531
0.242
We performed similar analysis for ATE es- timation on ER and SF datasets, on addi- tional CSuite datasets that contain discrete variables or are unidentifiable, and CATE estimation on a subset of CSuite. See Ap-
T+L T4N
0.001
1.522
0.105
0.109
0.848
0.819
0.003
1.532
0.241
0.015
0.597
0.168
pendix and Appendix E.4 Also, we test the case with latent confounders in Appendix E.7
On the semi-synthetic benchmark datasets, Twins and THDP, we evaluated both ATE and CATE estimation as shown in Figure 5 For ATE estimation, DECI-Spline is fractionally better than baselines Twins and significantly better for IHDP. On IHDP; it appears that only DECI-Spline was successful at ATE estimation. Given the right graph (Table 3 DECI-Spline is the best method for computing ATE. For CATE estimation, a similar pattern emerges. Additionaly, DECI-Spline have the similar ATE crror as True Graph + DECI Spline (Table for IHDP, implying it is also successful at discovering the graph.
Summary Across all experiments we see that DECI enables end-to-end causal inference with competitive performance on both synthetic and more realistic data. DECI particularly performs well compared to other methods when its ability to handle nonlinear functional relationship and non-Gaussian noise distributions comes into play in causal discovery or causal inference. Other ECI method combinations can achieve strong performance, but have weak performance if either step's assumptions are violated. We find DECI-Spline particularly attractive given its high degree of flexibility..
5 Discussion, scope, and limitations
Causal inference requires causal assumptions on the relationships between variables of interest. The field of causal discovery aims learn about these relationships from observational data, given some non-causal assumptions on the data generating process. Motivated by real-world application where our knowledge of causal relationships is incomplete, DECT combines ideas from causal discovery and inference to go directly from observations to causal predictions. This formulation requires us to adopt assumptions, namely, that the data is generated with a non-linear ANM and that there are no unobserved confounders. Empirically, we find DECI to perform well when these assumptions are satisfied, validating the viability of an end-to-end approach. However, the non-linear ANM assumptions made by DECT are impossible to check in most real- world scenarios. Thus, combining the output of discovery methods with incomplete causal assumptions an attractive avenue to make end-to-end methods more robust in the future. Interestingly, we also test the sce- narios where DECI's assumptions are violated. Specifically under 1. dataset with missingness (section 4.1 2. dataset with discrete variables (section 4.2) and 3. under latent confounders (appendix E.7]. we do not find its performance to degrade severely apart from the lalent confounder scenario. This encouraging result motivates us to extend our theoretical analysis to the mixed data type; missing data settings and improving the robustness under latent confounders future work. Finally, in thooretical considerations, certain im- portant aspects of the end-to-end method remain to be developed, including convergence rate, asymptotic optimality, and dealing with limited overlap and imbalance.
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References
Douglas Almond, Kenneth Y Chay, and David S Lee. The costs of low birth weight. The Quarterly Journal of Economics, 120(3):1031-1083, 2005.
Stoen A Andersson, David Madigan, and Michael D Perlman. A characterization of markov equivalence classes for acyclic digraphs. The Annals of Slolistics, 25(2)5505 541, 1997.
Joshua D Angrist, Guido W Imbens, and Donald B Rubin. Identification of causal effects using instrumental variables. Journal of the American statistical Association, 91(434):414445 1996.
Yashas Annadani, Jonas Rothfuss, Alexandre Lacoste, Nino Scherrer, Anirudh Goyal, Yoshua Bengio, and Stefan Bauer. Variational causal networks: Approximate bayesian inference over causal structures. arXiv preprint Xie:22100 07635, 2021.
Keith Battocchi, Eleanor Dillon, Maggie Hei, Greg Lewis, Miruna Oprescu, and Vasilis Syrgkanis. Estimating the long-term effects of novel treatments. arXiv preprint arXiv:2103 203.:9200..0899, 2021.
Rohit Bhattacharya, Tushar Nagarajan, Daniel Malinsky, and llya Shpitser. Differentiable causal discovery under unmeasured confounding. In International Conference on Artificial Intellignnce and Stalistics, pp- 2314 2322. PMLR, 2021.
Ioana Bica, Ahmed M Alaa, James Jordon, and Mihaela van der Schaar. Estimating counterfactual treat- ment outcomes over time through adversarially balanced representations. In International Conference on Learning Represenlations, 2019.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the Americon statistical Association, 112((18):8599-88 859-8777 2017.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double /dbiasd machine learning for treatment. and structural parameters. The Econometrics Journal, 21(1):C1 C6L, 2018.
David Maxwell Chickering. Optimal structure identification with greedy search. Journal of machine learning research, (NNov):577-554 2002.
David Maxwell Chickering and Christopher Meek. Selective greedy equivalence scarch: Finding optimal bayesian networks using a polynomial number of score evaluations. ar preprint arXiv:1506 021 13, 2015.
Max Chickering. Statistically efficient greedy equivalence search. In Conference Uncertainty in Artificial Intelligence, pp. 241-249. PMI.R, 2020.
Tom Claassen and Tom Heskes. A bayesian approach to constraint based causal inference. arXiv preprint arXiv:1210.4866 2012.
Chris Cundy, Aditya Grover, and Stefano Ermon. Bed nets: Scalable variational approaches for bayesian causal discovery. Advances in Neural Information Processing Systems, 34:7095-7110, 2021.
Tristan Deleu, António Göis, Chris Emezue, Mansi Rankawat, Simon Lacoste-Julien, Stefan Bauer, and Yoshua Bengio. Bayesian structure learning with generative flow networks. arXiv preprint arXiv:2202.1900 2022.
Conor Durkan, Artur Bekasov, Jain Murray, and George Papamakarios. Neural spline flows. In H. Wal- lach, H. Larochelle, A. Beygelzimer, F. d' A'Achhé-Bue, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL httpss://proceedinggs .ddf.
Nir Friedman and Daphne Koller. Being bayesian about network structure. a bayesian approach to structure discovery in bayesian networks. Machine learning, 50(1):95-1255 2003.
13

---

Published in Transactions on Machine Learning Research (06/2024)
Erdun Gao, Ignavier Ng. Mingming Gong, Li Shen, Wei Huang, Tongliang Liu, Kun Zhang, and Howard Bondell. Missdag: Causal discovery in the presence of missing data with continuous additive noise models. Advances in Neural Information Processing Systems, 35:5024 5038, 2022.
Clark Glymour, Kun Zhang, and Peter Spirtes. Review of causal discovery methods based on graphical models. Frontiers in genetics, 10:524, 2019.
Wenbo Gong, Kaibo Zhang, Yingzhen Li, and José Miguel HernándezzLLoatoo Active slices for sliced stein discrepancy, In International Conference on Machine Learning, pp. 3766-3776. PMLR, 2021.
Richard Guo and Emilija Perkovic. Minimal enumeration of all possible total effects in a markov equivalence class. In International Conference on Artificial Intelligence and Statistics, pp. 2395-2403. PMLR, 2021.
David Heckerman, Christopher Meek, and Gregory Cooper. A bayesian approach to causal discovery. Com- putation, causation, and discovery, 19:141-166, 1999.
David Hockorman, Christopher Mock, and Gregory Cooper. A bayesian approach to causal discovery. Inno- valions in Machine Learning: Theory and Applications, pp. 1-28, 2006.
Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217-244) 2011.
Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approximators. Newral networks, 2(5):359 366, 1989.
Patrik O. Hoyer, Dominik Janzing, Joris M. Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlincar causal discovery with additive noise models. In Daphne Koller, Dale Schuurmans, Yoshua Bengio, and Léon Bottou (eds.), Advances in Newral Information Processing Systems 21, Proceedings of the Twenty--Scond Annwal Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 8-11, 2008, pp. 689-696. Curran Associates, Inc., 2008a. URL http:/pss//rrroeeng...o. http:////0000/aae/////6000ceeae0
httml
Patrik O Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, Bernhard Schölkopf, et al. Nonlinear causal discovery with additive noise models. In NIPS, volume 21, pp. 689 696. Citeseer, 2008b.
Biwei Huang. Diagnosis of autism spectrum disorder by causal influence strength learned from resting-state fmri data. In Newral Engineering Techniques for Autism Spectrum Disonder, pp. 237-267. Elsevier, 2021.
Biwei Huang, Kun Zhang, Yizhu Lin, Bernhard Schölkopf, and Clark Glymour. Generalized score functions for causal discovery. In Proceedings of the 24th ACM SIGKDD International Conference Knowledge Discovery & Data Mining, pp. 1551-1560, 2018a.
Chin-Wei Huang, David Krueger, Alexandre Lacoste, and Aaron Courville. Neural autoregressive flows. International Conference on Machine Learning, pp. 2078 2087. PMLR, 2018b.
Aapo Hyvärinen and Stephen M Smith. Pairwise likelihood ratios for estimation of non-gaussian structural equation models. Journal of Machine Learning Research, 14(Jan):111-152 2013.
Guido W Imbens. Polential outcome and directed acyclic graph approaches to causality: Relevance for empirical practice in economics. Journal of Economic Literature, 58(4):1129-77 79, 2020.
Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, 2015.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144 2016.
Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul. An introduction variational methods for graphical models. Machine learning, 37(2):183-25) 233, 1999.
14

---

Published in Transactions on Machine Learning Research (06/2024)
Yonghan Jung, Jin Tian, and Elias Barcinboim. Learning causal effects via weighted empirical risk mini- mization. Advances in neural information processing systems, 33, 2020.
Youghan Jung, Jin Tian, and Elias Bareinboim. Estimating identifiable causal effects on markov equivalence class through double machine learning. In Marina Mcila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 5168 5179. PMLR, 18 24 Jul 2021. URL htttps://proceeelnggs mll. wtwps//139////eeee /
jung21b. http///119v///gnb...oo html
Marcus Kaiser and Maksim Sipos. Unsuitability of NOTEARS for causal graph discovery. CoRR, abs/2104.05441,, 2021. URL http:://arr...orgaabs///// 05441
Markus Kalisch and Peter Bühlman. Estimating high-dimensional directed acyclic graphs with the pc- algorithm. Journal of Machine Learning Research, 8(3), 2007.
llyes Khemakhem, Ricardo Monti, Robert Leech, and Aapo Hyvarinen. Causal autoregressive flows. In International Conference on Artificial Intelligence and Statistics, pp. 3520--528. PMLR, 2021.
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412 12. 6980, 2014.
arXiv preprint
Diederik P Kingma and Max Welling, Auto-encoding variational bayes. arXiv preprint drXiv:1312.6114, 2013.
Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Πlya Sutskover, and Max Welling. Improved variational inference with inverse autoregressive flow. Advances in neural information processing systems, 2016.
Jeremias Knoblauch. Frequentist. consistency of generalized variational inference. arXim:1912044440 2019.
arXio preprint
Sören R Künzel, Jasjeet S Sekhon, Peter Bickel, and Bin Yu. Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the national academy of sciences, 116(10):4156- 4165, 2019.
Sébastien Lachapelle, Philippe Brouillard, Tristan Deleu, and Simon Lacoste-Jllien. Gradient-bascl neural dag learning. Xr preprint ar 1906. 000602226, 2019.
Fan Li, Kari Lock Morgan, and Alan M Zaslavsky. Balancing covariates via propensity score weighting. Jowrnal of the American Statistical Association, (115211)390-4440 2018.
Phillip Lippe, Taco Cohen, and Efstratios Gavves. Efficient neural causal discovery without acyclicity constraints. arXie preprint drXi::2107.14483, 2021.
Po-Ling Loh and Peter Büthlmann. High-dimensional learning of linear causal networks via inverse covariance estimation. The Journal of Machine Learning Research, 15(1):3065 3105, 2014.
Lars Lorch, Jonas Rothfuss, Bernhard Schölkopf, and Andreas Krause. Dibs: Differentiable bayesian struc- ture learning. Advonces in Neural Information Processing Systems, 34:24111 24123, 2021.
Christos Louizos, Uri Shalit, Joris Mooij, David Sontag, Richard Zemel, and Max Welling, Causal effect inference with deep latintlv models. arXiv preprint arXie:1705.0880 2017.
Chao Ma, Sebastian Tschiatschek, Konstantina Palla, José Miguel Hernández-Lobato, Sebastian Nowozin, and Cheng Zhang. Eddi: Efficient dynamic discovery of high-value information with partial vae. arXiv preprint XrXiv:1809.111 11142, 2018
Chris J Maddison, Andriy Mnih, and Yee Whye Teh. The conerete distribution: A continuous relaxation of discrete random variables. Xiv preprint arxio: 1xXx:1611.000122 2016.
15

---

Published in Transactions on Machine Learning Research (06/2024)
Takashi Nicholas Maeda and Shohei Shimizu. Red: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In International Conference on Arlificial Intelligence and Statistics, pp. 735 745. PMLR, 2020.
Takashi Nicholas Maeda and Shohei Shimizu. Causal additive models with unobserved variables. In Uncer- tainty in Artificial Intelligence, pp. 97 106. PMLR, 2021.
Keith A Markus. Causal effects and counterfactual conditionals: contrasting rubin, lewis and pearl. Eco- nomics & Philosophy, 4713)144 2021.
Kevin Murphy. Active learning of causal bayes net structure. Technical report, technical report, UC Berkeley, 2001.
Radford M Neal et al. Meme using hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.
AS Nemirovsky. Optimization ii numerical methods for nonlincar continuous optimization. 1999.
Ignavier Ng. Shengyu Zhu, Zhitang Chen, and Zhuangyan Fang. A graph autoencoder approach to causal structure learning, ar preprint or orXiv:1911.07420, 2019.
Ignavier Ng, AmirEmad Ghassami, and Kun Zhang. On the role of sparsity and dag constraints for learning linear dags. XrXiv preprinl arXiv:2006. 10201, 2020.
Nick Pawlowski, Daniel Coelho de Castro, and Ben Glocker. Deep structural causal models for tractable counterfactual inference. Aduonces in neural information processing systems, 33:857-869, 2020.
Pearl. Causal diagrams and the identification of causal effects. Causality: Models, Reasoning, and Inference. Cambridge University Press: New York, 2000.
J. Pearl. The logic of structure-based counterfactuals. Causality: Models, Reasoning and Inference, pp.
201 259, 2009a.
Judea Pearl. Causal inference in statistics: An overview. Statistics surveys, 3:96-146, 2009b.
Judea Pearl. Causality. Cambridge university press, 2009c.
Jonas Peters and Peter Bühhlman. Identifiability of gaussian structural equation models with equal error variances. Biometrika, 101(1):219-22) 2014.
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithmss. The MIT Press, 2017.
Alexander G. Reisach, Christof Seiler, and Sebastian Weichwald. Beware of the simulated dag! varsortability in additive noise models. CoRR, abs abx/2200.136617, 2021. URL htttps///axxiv org/abs/2102 13647
Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International con- ference machine learning, pp. 1530-1538. PMLR, 2015.
Paul Rosenbaum and Donald B Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41-55 11,5,, 1983.
Donald B Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 6685) 1974.
Donald B Rubin. Inference and missing data. Biometrika, 63(3):581-59)) 1976.
Karen Sachs, Omar Perez, Dana Pe'er, Douglas A Lauffenburger, and Garry P Nolan. Causal protein- signaling networks derived from multiparameler single-cell data. Science, 308(522)):523 029,, 2005.
Amit Sharma, Vasilis Syrgkanis, Cheng Zhang, and Emre Kıeman. Dowhy: Addressing challenges in expressing and validating causal assumptions. aixiv preprint ar Xix:2100.. 13518, 2021.
16

---

Published in Transactions on Machine Learning Research (06/2024)
Shohei Shimizu, Patrik O Hoyer Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non- gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(10), 2006.
Peter Spirtes and Clark Glymour. An algorithm for fast recovery of sparse causal graphs. Social science computer review, 9(1):62-72, 1991.
Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.
Daniel J Stekhoven and Peter Bühhlannn Mis-forest--nn--araaae type data. Bioinformatics, (8(1):112-113) 2012.
non hon-maaaettric missing value imputation for mixed-
Eric V Strobl, Shyam Visweswaran, and Peter L Spirtes. Fast causal inference with non-random missingness by test-wise deletion. International journal of data science and analytics, 6(1):47 62, 2018.
Elizabeth A Stuart. Matching methods for causal inference: A review and a look forward. Statistical science: a review journal of the Institute of Mathematical Statistics, 25(1):1, 2010.
Simon Tong and Daphne Koller. Active learning for structure in bayesian networks. In International joint conference artificial intelligence, volume 17, pp. 863-861 869. Citeseer, 2001.
Ruibo Tu, Cheng Zhang, Paul Ackermann, Karthika Mohan, Hedvig Kjellström, and Kun Zhang, Causal discovery in the presence of missing data. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 1762 1770. PMLR, 2019.
Tim Van den Bulcke, Koenraad Van Leomput, Bart Naudts, Pict van Remortel, Hongwu Ma, Alain Ver- schoren, Bart De Moor, and Kathleen Marchal. Syntren: generator of synthetic gene expression data for design and analysis of structure learning algorithms. BMC bioinformatics, 7(1):1-12, 2006.
Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, (13(523):1228-12) 2018.
Yixin Wang and David Blei. Variational bayes under model misspecification. Advances in Neural Information Processing Systems, 32, 2019.
Yixin Wang and David Blei. Frequentist consistency of variational bayes. Journal of the American Statistical Association, 2018
Felix Xinnan X Yu, Ananda Theertha Suresh, Krzysztof M Choromanski, Daniel N Holtmann- Rice, and Sanjiv Kumar. Orthogonal random features. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett (eds.). Advances in Neural Information Processing Systems, vol- ume 29. Curran Associates, Inc., 2016. URL https://prooceednng: www.seenrrrrrroooooe.oe ////pp/s/c//pppr////////////// 0.adda449444000777997346666622
Paper.
Cheng Zhang, Judith Bütepage, Hedvig Kjellstrom, and Stephan Mandt. Advances in variational inference. IEEE transactions on pattern analysis and machine intelligence, 41(8):2008-20) 20000,22226,220 2018.
K ZHANG. On the identifiability of the post-nonlincar causal model. In Proceedings of the 25th Conference on Uncerlainly in Artificial Intelligence (UAI), volume 647. AUAI Press, 2009.
Keli Zhang, Shengyu Zhu, Marcus Kalander, Ignavier Ng, Junjian Ye, Zhitang Chen, and Lujia Pan. geastle: A python toolbox for causal discovery. arXio preprint orX1v:2111.1515,, 2021.
Kun Zhang and Lai-Wan Chan. Extensions of ica for causality discovery in the hong kong stock market. Tn International Conference on Neural Information Processing, pp. 400 409. Springer, 2006.
Kun Zhang, Zhikun Wang, Jiji Zhang, and Bernhard Schölkopf. On estimation of functional causal models: general results and application the post-nonlinear causal model. ACM Transactions on Intelligent Systems and Technology (TIST), 7(2):1 122, 2015.
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Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P Xing. Dags with no tears: Continuous opti- mization for structure learning. arXiv preprint 1r:::11:8.0000 002,0 2018a.
Xun Zheng, Bryon Aragam, Pradeep K Ravikumar, and Eric P Xing. Dags with no tears: Contin- uous optimization for structure learning. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (ods.), Advances in Neural Information Processing Systems, vol- une 31. Curran Associales, Inc., 2018b. URL https://proceeinns..ee G3fro1111FFF00000000000000101000 pdff
Xun Zheng, Chen Dan, Bryon Aragam, Pradeep Ravikumar, and Eric Xing. Learning sparse nonparametric dags. In International Conference on Artificial Intelligence and Statistics, pp. 3414-3425. PMLR, 2020.
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A Theoretical Considerations for DECI
DECI can be categorized as a functional score-based causal discovery approach, which aims to find the model parameters 0 and mean-field posterior qo(G) by maximizing the ELBO (eq. (9). A key statistical property of DECI is whether it is capable of recovering the ground truth data generating distribution and truc graph Go when DECT is correctly specified and with infinite data. In the following, we will show that DECI is indeed capable of this under standard assumptions. The main idea is to first show that the maximum likelihood estimate (MT.E) recovers the ground truth due to the correctly specified model. Then, we prove that optimal solutions from maximizing the ELBO are closely related the MLE under standard assumptions.
Theorem (DECI recovers the true data generating process). Assume DECT's underlying SEM belongs to continuous additive noise structural equation models with (1) non-invertible, 3rd-order differentiable function between variables that are not constant w.r.t. any of their inputs (t.e. non-trivial); (2) differentiable noise densily und bounded log-likelihood, then, under correclly specified model, infovile dala limit and no lalent confounders in data generating mechanism, the solution (6', (g(G)) from marimizing the ELBO (eg. (9)) satisfies (G) 8(G = G') where G' is a unique graph. In particular, G' G and po (x: G') ) p(x: G
Our proof strategy is as follows. In Appendix A..1] we show that the assumptions of the theorem imply certain well-known properties and standard causal assumptions. In Appendix [A.2] we prove that maximum likelihood cstimation identifies the correct data generating process, and in Appendix A.3 we complete the proof by showing that maximizing the ELBO asymptolically recovers the MLE. Appendix A.4 discusses additional theoretical results for treatment effect estimation.
A.1 Notation, Assumptions and definitions
First, we define the notation and explain the implication of assumptions required for our theory. We denote random variable RD with a ground truth data generating distribution p(x; G"), where G is a binary adjacency matrix representing the true causal DAG. DECT is an additive noise model (ANM), defining the structural equations zj = (Xpa(t:))):) 0) + 2j, where pa(j; G)) are the parents of node j specified by the adjacency matrix G, f are non-invertible, third-order differentiable non-trivial functions and 2, are mutually independent noise variables with a joint density po(21...., 2D)- The mean-field variational distribution go (G( is product of independent Bernoulli distribution, and p(G) is the soft prior over the graph defined by eq. (5)
For completeness, we first provide the definition of causal minimality, structural identifiability, and explain our model specification and regularity assumptions.
Definition 1 (Minimality). For a distribution ps(x: G) induced by SEM with graph G and parameter 0, we say the model satisfies the causal minimality condition if the distribution po (x: G) does not satisfy the local Markov condition with respect to any sub-graph of G.
Definition 2 (Structural Identifiability) For a distribution (x(x; G) induced from SEM, the graph G said be structural identifiable from po (x: G) if there crists no other distribution po (x:G') such that G + G' and po (x; G) = pa (x; G').
The following assumptions are explicit assumptions of Theorem 1 and are restated more formally now, along with a discussion of when such assumptions are likely to hold in practice.
Assumption (Correctly Specified Model). We assume the DECI model is correctly specified. Namely, there exisls a purameler 0* such that pa- (x; GD) = p(x; G
In practice, this assumption is hard to check in general. Assumption 2 (Regularity of log likelihood). We assume for all parameters 0 and possible graphs G, the following holds:
Ep(x,0o) || log (x C)]]
This assumption can be verified by analysing the tail behaviour of the noise distribution of the DECT SEM.
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We can now show that the our assumptions about the DECI model imply some of the commonly used causal assumptions.
Lemma 1 (Implication of DECI assumptions). Assume DECI satisfies the assumptions mentioned in The-
Then DECI satisfies (1) the causal Markov assumption; (2) causal minimality; (3) structural identifiability.
orem
1
Proof. We show how each assumptions are implied by referring to some theorems from provious works.
Causal Markov assumption. Since DECI belongs to the class of SEMs with differentiable probability density functions, from Theorem 6.22 in Peters et al. (0117) the global, local and Markov factorization properties are equivalent. This also allows the gradient-based optimization to be used for the induced SEM. Theorem 1.4.1 in Pearl (2009ch proves that the induced joint probability from SEM is Markovian with respect to the graph. We typically call this the causal Markov property.
Causal minimality. Since we assume the induced distribution from DECI SEM has a density and it is Markovian to its graph, Proposition 6.36 in Peters et al. (2017) proves that if the child node is not independent to any of its parent nodes conditioned on other remaining parents, then satisfies the causal minimality. Since we assume that the functional relations wetween output and inputs are not constant, they cannot be independent. Thus, by Proposition 6.36, it satisfies causal minimality.
Structural identifiability. From the Proposition 23 in Peters & Bühlmann (014)), the necessary condi- tions for nonidentifiability require the functions to be either linear or monotonic. However, we assume DECI functional relationship is non-invertible, which violates these conditions. Since we assume causal minimality for DECI SEM, from Theorem 20 in Peters & Bühlmann (2014), DECI SEM structural identifiable.
Comparisons to MEC-based approaches DECI aims to recover the full causal structures by restricting the relatioshppp as ANM. Alternative to this, there exists MEC- approaches that can identify
the causal quantities upto a Markov equivalence class (Guo & Perkovic 2021 Jung al. 2020). Guo &
Perkovic (2021); Jung et al. (2020) requires the access to the partial ancestral graph (PAG) to start with, and can only work with iet-identifaaiility For example, Guo & Perkovic (2021) enumerates all possible effects associated to the equivalence class. On the other hand, DECI also enumerates all the causal effects associated to the posterior distribution of the cansal graph. One limitation of the above approaches is that they cannot handle latent confounders. In addition, althogh the PAG required by Guo & Perkovic (2021 ; Jung et al.) (2020) can be learned by conditional independence (CI) test, making use of them is non-trivia in practice. Maeda & Shimizu (2020 2021); ); Bhattacharya et al. (2021) showed that CI-based approach
underperform the functional- sanstional-based and differentiable methods, which is the foundations of DECI.
A.2 MLE Recovers Ground Truth
The likelihood has often been used the score function for causal discovery. For example, Careft (Khe- makhem et al. 2021) adopts the likelihood ratio test (Hyvärinen & Smith 2013) in the bivariate case, which is equivalent to selecting the causal directions with the maximized likelihood. However, they did not explic- itly show that the resulting model recovers the ground truth for the multivariate case. In addition, [Zhang et al. 2015) proved that maximizing likelihood for bivariate causal discovery is equivalent to minimizing the dependence between the cause and the noise variable. With the correctly specified, structural identifiable model, the resulting noise and cause are independent through maximizing the likelihood, indicating the graph is indeed causal. However, it is non-trivial to generalize this to the multivariate case that we treat in DECL. In the following, we will show that under correctly specified model and with maximum likelihood training with infinite data, DECI can recover the unique ground truth graph G* = Go and the truc data generating distribution pg px; G*) p(x; G"), where (0", G*) are MLE solutions.
Proposition 1. Assuming assumptions in Theorem 7 are salisfied, we denole (²) as the MLE sobution with infinite training data. Then, we have
po- (x: G*) = p(x; Go)
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In porticular, we have G G
Proof. The key idea is to show that with arbitrary (0, G), we have the following:
N
N
lim N
log po (x;;; G) G) S
lim N
log p(x;; G
N
By law of large numbers, we have
N
lim No00
log po (x;; G) = R P() (1o G))
N
Then, we can show
Bp(x:00)
log (x G)]
E p(x; px: G*)]
log (x: G)
- -
p(x; G*)
(x: G)
-1
/ po G)dx
p(x;
where the inequality is due to logt < t 1 1. With assumption 3-4, we know there latent confounders and the model is correctly specified. Then, the above equality holds when (0", G*) induces the same join likelihood p(x; G Since the model is structural identifialle, we must have G = GO.
A.3 DECI Recovers the Ground Truth
To show that DECT can indeed recover the ground truth by maximizing the ELBO, we first introduce important lemma showing the KL regularizer [LIP(()) is negligible in the infinite data limit.
Lemma Assume variational distribution (²(G) over a space of graphs Gø, where each graph G Go has non-zero associated weight Wy(G). With the soft prior p(G) defined as eq. (5) and bounded X, p. a, have
N ([(a))))M -. N
lim
(15)
Proof. First, write down the definition of KL divergence
(L)gg())(lp(G)) = Σ (G) [log (G) + A||G|| + ph(G)² + +a((G) + log 2]
GES
where is the normalizing constant for the soft prior. From the definition and assumptions, it trivial to know that log (G), 2|| are bounded for all G Gg. In the following, we show that h(G) and log are also bounded.
From the definition of the DAG penalty, we have h(G) tr(exp o G)) D. The matrix exponential is defined as
tr(exp(G @ G)) =
tr((G O
k!
0=0
k! tr((G)+)
km
0
(r(())
K
k=
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where the second equality is due to the fact that is binary adjacency matrix. From Zheng et al. (2018b). we know that tr(G*) counts for the number of closed loops with length k. Since the graph has finite number of nodes, the longest possible closed loop is D, resulting in the third equality.
Thus, it is obvious that for any k, the number of closed loops with length must finite. Hence, is trivial that h(G) oc. Therefore, with bounded À; p. a, the un-normalized soft prior
exp(->N|G||> ph()) - ah(G)) < 00.
Thus, the normalizing constant Z must be finite since there only finite number of possible graphs Therefore, these must exists constant Ma.c such that log u (b(()) +G))+ +g (ahli Z < Ma.G. Hence, we have
S (L(go(G)llp())
< Σ
wo(G) v V
E GEG
wg(G)
Σ
M < 20.
GEGs
GEGO
Thus, have
lim ([III() -0 N00 N
where the third incquality is obtained by using Cauchy-Schwarz incquality.
Now, we can prove Theorem T
Proof. In terms of optimization, it equivalent to re-write the ELBO (eq.
9 as
1 Eys [xo XN [w//) - N KL (G)
KL
Now, under the infinite data limit and the definition of yo, we have
lim
Egg [
XN|G)]
KL (G)(G)) N
N 0c
lim
Σ
N (Llas
N toc
GED
lim
wo(G) log
Noc
i=1 GEGs
= =ʃ
=fp(x: p(x: GO)
Σ GEG
Wo(G) log po (x G)dxx
where the second and third equalities are from Lemma and the law of large numbers, respectively. Let (0", G*) be the solutions from MLE (Proposition D.. Then, we have
s (lgg pa- (x; G*)]
for all graph Go due to the MLE solution (G*,0"). Then, since Eces, wa (C) = 1, w& > O, we have
Σ
(G))Epp()
[log [log < (9) [log pa. (x: G*)]
GEG
with the equality holding when every graph G E G and associated parameter lg satisfies
Ep(x;99) [log pa. (x[G)) = Exxx;99 [log pg- (x|G*)]
(16)
Note that when eq. 16 is satified, the ELBO is maximized based on its definition. From proposition under correctly specified model, we have
E Ep(x;09) [log (x(G*)) Gp) [lg G*)]
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Thus, for G' E Gg and associated parameter 0', the condition in eq.
16 becomes
Bp(x;G") [log po pa (x|G")] E Gxxx;G9) [[og((x|G)]
B Bx(x)00)) log (x; G') 0
pxx; GO)
[T|[p(x; G Ilpo G²)
which implies po (x: G") p(x; GO). Since DECI is structural identifiable, this means G' G and it is unique. Thus, the graph space 96 only contains one graph G", and (G( = G').
One should note that we do not explicitly restrict the noise distribution, indicating it still holds with the spline noise (Section 3.4h. However, the above theorem implicitly assumes that DECI a special case of ANM for structural indentifiability and that the data has no missing values. Thus, is not applicable for DECI with the mixed-type and missing value extensions. We leave more general theoretical guarantee to future work.
A.4 Theoretical considerations for treatment effect estimation
Our proposed DECI model provides an end-to-end framework to connect causal discovery and inference. The assumptions introduced in Appendix [A.] are mainly introduced for the causal discovery lask. In the following, we will discuss their relationships to some common assumptions required for the causal inference task. In particular, we show that no additional assumptions are needed to guarantee that DECI can correctly estimate treatment effects.
Causal effect identifiability and positivity. Under the assumed DECI SEM (non-linear additive noise model with independent noise variables), our model is Markovian, meaning that the well-known causal effect identification theorem for Markovian models naturally holds (Theorem 3.2.5 of Pearl (2000 ). This result does assume positivity of observational distributions.
Fortunately, the proposed form of DECI implies positivity. In particular, DEC1 is based on non-linear ad- ditive noise models. One of the fundamental assumption of non-linear ANMs is that the joint distribution should have a proper density function, i.e. it should be absolutely continuous w.r.t. Lebesgue measure. Addi- tionally, we assume DECI has differentiable density function, so the density is always positive. Additionally, since the noise model for DECI is either a Gaussian or a spline transformed Gaussian, the measure of the set where the density is zero must have a measure zero. This means DECI will have a positive density almost everywhere, which in turn implics positivity.
In nutshell, one the DECI is learned with assumption and 2, the causal effect associated to the underlying SEM is always identifiable. The remaining question is whether the identified causal effect is the correct estimate of the true effect. This requires the DECT recovers the underlying causal system which is guaranteed by Theorem 1. On potential limitation is that assumption 1,2 and global optimization are hard to verify, meaning that the estimated causal effect may not be the truth.
A note on consistency assumptions. The consistency assumption is not required by our method, as these are typically used only within the potential outcome formulation for causality of Rubin (Rubin) 1974]. and are rarely seen in graphical model based approach (Pearl 2000) to causality adopted in our paper. The graphical model approach follows a different philosophy defined causality: we start by defining causal model, and all other counterfactual and interventional quantitics are derived from this causal model. On the contrary, the PO approach treats counterfacluale (i.e. polential outcomes) as primitive random variables that are not observed. Therefore, the PO approach needs assumptions such as consistency, no interference, etc. to ensure the coherent mathematical consistency of counterfactual statements. Soe Imbens) (2020): Pearl (2009a); Markus (2021), for example, for more discussions on PO approach versus graphical approach causality.
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Entropy of veriational adjacinoy distribution
100
10--
0.0
0.4
0.8
Training size
Figure 6: The entropy of the graph posterior 9s w.r.t. training dataset size.
A note on causal graph in causal inference For causal inference approaches, like TLearner, SLearner Künzzll et al. 2019). etc., also assume an implicit causal graph. In general, covariates act as confounders, and there is a direct connection from the treatment to the effect variable. However, how to pick the covariate variable challenging, and most approaches simply pick all remaining variables, leading to an incorrect graph in most cases. Additionally, strong prior knowledge is also required propose a reasonable graph, For example, it is hard in general to find all possible mediators between treatment and effect. However, DECI relieves this problem by incorporating a graph learning stage. Once trained, DECI can leverage the identified graph distribution to estimate the effect on the target variable. Therefore, only the relevant variables will be selected as covariates, and DECI can naturally deal with mediators.
Effect to causal estimation of incorrect graph with low probability The way DECI used to estimate causal quantitics is based on the Monte Carlo approximation due to the intractability of the expectation graph posterior. Therefore, the quality of the inference depends heavily the graph distribution and the number of samples used in the MC. If the incorrect graph has very low probability such that it is rarely sampled, then the estimation quality will not be affected with smaller graph sample size. But the variance of the quantity will be high. If the graph sample size increases towards infinity, the estimation variance will tend towards zero, but the incorrect graph will have innegligible effect, and result in biased estimator.
A.5 Convergence to single graph with large dataset size
Based on the theorem [ the graph posterior tends to collapse a single graph. We conduct additional experiment to demonstrate this trend. First, we sample a dataset similar the cxisting ones from ER with 16 nodes and 64 edges (section 4.1, with 1M training data points and 5000 in separate validation dataset. We then sub-sample the training dataset to contain additional datasets with 1000, 10, 000, 100, 000, 500, 000, 900, 000 points, where each smaller subset is fully contained in each larger one. Finally, we train DECT on those training sets with the same seed, adjusting the augmented Lagrangian parameters linearly with the training set size to ensure a consistent. number of training epochs are seen.
From these runs, we plot the entropy of the graph posterior with the training dataset size (Figure 6). We can clearly observe that the entropy decreases with the training data size, indicating its tendency of converging to single graph.
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B Additional Details for DECI
B.1 Optimization Details for Causal Discovery
As mentioned in the main text, we gradually increase the values of p and a as optimization proceeds, so that non-DAGs are heavily penalized. Inspired by Notears, we do this with a method that resembles the updates used by the augmented Lagrangian procedure for optimization (Nemirovsky 1999). The optimization process interleaves two steps: (i) Optimize the objective for fixed values of p and for certain number of steps: and (ii) Update the values of the penalty parameters & and The whole optimization process involves running the sequence (i)-(ii) until convergence, or until the maximum allowed number of optimization stops is reached.
Step (i). Optimizing the objective for some fixed values of p and a using Adam [Kingma Ba 2014p. We optimize the objective for a maxinum of 6000 steps or until convergence, whichever happens first (we stop early if the loss does not improve for 1500 optimization steps. If so, we move step (ii)). We use Adam, initialized with a step-size of 0.01. During training, reduce the step-size by factor of 10 if the training loss does not improve for 500 steps. We do this maximum of two times. If we reach the condition a third time, we do not decrease the step-size and assume optimization has converged, and move to step (ii).
Iterating (i)-(i).. We initialize = 1 and a = O. At the beginning of step (1) we mcasure the DAG penalty Pi = (G) (G). Then, we run step (i) as explained above. At the beginning of step (ii) we measure the DAG penalty again, P2 (G)) If P2 0.65 P1, we leave p unchanged and update + a + p P2. Otherwise, if P3 2 0.65 Pl, we leave unchanged and update p + p. We repeat the sequence (i) (ii) for maximum of 100 steps until convergence (measured as or p reaching some max value which we set to 10t3 for both), whichever happens first.
B.2 Other Hyperparameters.
We A, = 5 in our prior over graphs eq. 5 For ELBO MC gradients we use the Gumbel softmax method with a hard forward pass and soft backward pass with temperature of 0.25.
The functions eq. 7 used in DECT's SEM, and f, are 2 hidden layer MLPs with 128 hidden units per hidden layer. These MLPs use residual connections and layer-norm at every hidden layer.
For the non-Gaussian noise model in eq. the bijection x is an bin rational quadratic spline Durkan ct al.
2019 with learnt parameters.
In section 3.3 for ATE estimatiin we compute expectations by drawing 1000 graphs from DECT's graph posterior 4g and for each graph draw 2 samples of xy for total of 2000 samples. For CATE estimation, we need to train separate surrogate predictor per graph samples. We draw 10 different graph samples and 10000 (xc, xy) pair samples for each graph. We use these to train the surrogate models.
Our surrogate predictor is a basis function linear model with 3000 random Fourier features drawn such that the model approximates Gaussian process with a radial basis function kernel of lengthscale equal to 1 Yu et al. 2016).
B.3 ELBO Derivation
The goal of maximum likelihood involves maximizing the likelihood of the observed variables. For DECI (with fully observed datasets) this corresponds to the log-marginal likelihood
x logpo(x) "g..,e).....,z") = log Σ P Il (o
(17)
Marginalising in the equation above intractable, even for moderately low dimensions, since the num- ber of terms in the sum grows exponentially with the size of G (which grows quadratically with the data dimensionality D).
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Variational inference proposes to distribution go(G) to build the ELBO, lower bound of the objective from eq. 17 as follows:
log po ,...,,,N) log Σ ( II po(""
(18)
p ()) II, Po(x" (G) (G)
log Σ (G)
(19)
qó (G)
((G II, Po po G)
llgg (v)(G)
(20)
[+(G)
log p(G) II, (o( G)
(c)
(Jensen's inequality)
(21)
(G)
(G)
log logp(C ((G)
Il Po Po(x"I)) n
+ (g+))
(22)
ELBO(
(23)
where we denote Hgg+) (G) log qo (G) for the entropy of the distribution ds- Interestingly, the dis- tribution de that maximizes the ELBO is exactly the one that minimizes the Kl-divergence between the approximation and the true posterior, (I(gs(G)llpaGIH) v). (see, c.g. Blei et al. (017)). This is why as can be used as a posterior approximation.
B.4 Intervened Density Estimation with DECI
Apart from (C)ATE estimation, DECI may also be used to evaluate densities under intervened distributions. For a given graph, the density of some observation vector a is computed by evaluating the base distribution density after inverting the SEM
Po = ax=aaGr a = II M(. (a; - Ji [i(apa(tG=))))
(24)
noting that the transformation Jacobian is the identity. We then marginalise the graphs using Monte Carlo:
M
po a) 2
po = a GD ~ (G(
(25)
M
In the rest of this section we derive methods that allow using DECI to estimate causal quantities.
Under Gāo(xz); i E T correspond to parent nodes and we have the following factorisation:
p(x:r Gdo((x)) Herr p(x:). We can then evaluate the interventional density of an observation Xyr = a with DECI as
= aldooxx7 b), Gm Po(X a, a, b| b|Grn(xT)
po (XT b) (lGr))
po(X = a|xi = b. du((x)) (xo(X = b)
pe (x = b)
II JEYT
p(z, (a, f((
(26)
which amounts evaluating the density of the exogenous noise correspondint non-intervened variables. We can then marginalise the graph using Monte Carlo as in eq. 25
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B.5
Relationship with Khemakhem et al. (2021)
Khemakhem et al. 2021 introduced Carefl method that uses autoregressive flows (Huang et al. 2018b; Kingma et al. 2016 to learn causal-aware models, using the variables' causal ordering to define the autore- gressive transformations. The method's main benefit is its ability to model complex nonlinear relationships between variables. However, Careft alone is insufficient for causal discovery, as it requires the causal graph structure as an input. The authors propose a two-step approach. First, run a traditional constraint-basol method (e.g,, PC) to find the graph's skeleton and orient as many edges as possible, and second, fit several flow models to determine the orientation of the remaining edges. The drawbacks of this approach include an external causal discovery methods (which will inherently limit. Carefl's performance
the dependence to that of the method used), and the cost of fitting multiple flow models to orient the edges that are left unoriented after the first step. Our method extends Khemakhen et al. 2021 to learn the causal graph among multiple variables and perform end-to-end causal inference.
B.6 Discussion on Causal Discovery Methods
When performing causal discovery, DECT returns a posterior over graphs. Most other causal discovery methods return either a single graph or an equivalence class of graphs. However, we can re-cast these methods in the probabilistic framework used by DECI by noting that posterior over graphs takes the form
p(X(G)p(G)
PG = EG ΣGP(XIG)P(G)
(27)
In this equation, the likelihood measures the degree of compatibility of certain DAG architecture with the observed data. For score- based discovery methods (Chickering 2002 Chickering & Meek 2015 Chickering,
2020 Huang et al. 2018a) we take the score to be log p( ((G). For functional discovery methods (Hoyer
itt al. 2008b Shimizu et al., 2006; ZHANG 2009) we use the exogenous variable log-density. Constraint- based methods (Spirles & Glymour 1991 Spirtes et al.] 2000) can also be cast this light by assuming a
uniform distribution over all graphs their outputted equivalence class G: log lxg((XX)) = =llog [9), E G. To what degree these methods succeed at constraining the space of possible graphs will depend on how well their respective assumptions are met and the amount of data available (Hoyer et al. 2008a).
B.7 Learning from interventional dataset
In this section, discuss how to extend our method to enable learning from interventional dataset. Assuming we have gathered S different interventional datasets D, = {x [ 8 1, 1,..., S from the same causal system; and for each D. it is associated with set of indiccs of interventional variables, T(a)- For pure observational dataset, assume T(x) Then, the training objective becomes:
ELBO(O, = B logp(C) G)) II (x (x Dy
+ H(9))
(28)
where Gao(x
denotes the mutilated graph under intervention X
When S = and T(1) =;
Equation
28 reduces to the fully observational case, Equation 9
C
Unified View of Causal Discovery Methods
This section introduces simple analysis showing that, similarly to DECI, most causal discovery methods based on continuous optimization can be framed from a probabilistic perspective fitting a flow. The benefits of this unified perspective are twofold. First, it allows a simple comparison between methods, shedding light on the different assumptions used by each one, their benefits and drawbacks. Second, it simplifies the development of new tools to improve these methods, since any improvements one of them can be easily mapped to the others by framing them in this unified framework (e.g. our extensions handle missing values and flexible noise distributions be easily integrated with Notears).
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The connection between causal discovery methods based on contimuous optimization and llweb models uses the concept of weighted adjacency matrix ( E DxD linked to function f(x; 0) RDD + RRD. Loosely speaking, these matrices can be seen as characterizing how likely is each output of f(x; 0) to depend on each component the input x. For instance, 5 = indicates that fi 8) is completely independent of z.y. Such adjacency matrices can be constructed efficiently for a wide range of parameterizations for J, such as multi layer perceptrons and weighted combinations of nonlinear functions. We refer the reader Zheng al. (2020) for details. In the following, lemma 4 shows that the objective function of fitting the flow is equivalent to solving the maximum likelihood problem of the noise distribution. Therefore, this allows the objective functions in many existing causal discovery methods to be unified under the maximum likelihood perspective.
Lemma 3. Let G represents a binary adjacency matrix, fa : TR Rd function whose ith output only depends on the parents of x; specified in G, and (x(x) be the Jacobian of go(x) x- fx(x). IfG represents DAG, then det Ja [x(x
Proof. We split the proof in several simple steps.
go (x) has a Jacobian-determinant of de - Jf), where Jy the Jacobian of fa(x).
J has non-zero entries exactly in the positions where GT non-zero. Therefore, it remains the DAG structure.
3. Matrix with DAG structure is nilpotent (i.c. all eigenvalues zero). Thus, Jy can be factorized into QUQ* where Q is unitary and U is strictly upper triangular (Schur factorization).
4. Finally, det(T Jf) = det(T - QUQ*) = det - U) =1
Lemma 4. Let f(x;0): RD RDD be 0-parameterized function with weighted adjacency matrir ( E R DDDD Given dataset ".......,,,, fitting a flow with the transformation fxx 8), base distribution Pu and hard acyclicily construint W (0) is equivalent to solving
max 6
logPa(x" 0)) s.t. A(W(0)) = 0.
(29)
where 1(-) is the algebraic characterization of DAGs from Zheng et al. 2018a
Proof. The acyclicity constraint enforced by constraining the optimization domain to θ = {0: h(W 0}. Then, the maximum likelihood objective can be written as
=
Σ log (x" f(x";0)) log det d(x" s(x";
log pog"o(x")
(30)
dx
= Σ logp: (x" f(x;;0))
(31)
where the first equality we use the change of variable formula, valid because the transformation z = g(x; 0)
x f(x; 0) is invertible for any f O, and the second equality uses that the function x" f(x";6) has Jacobian-determinant equal to (lemma 3 due to the constraint θ (0: : h(W()) 0).
In summary, recently proposed causal discovery methods based continuous optimization can be formulated from probabilistic perspective fitting flow with different constraints, transformations, and base distri- butions. This unified formulation sheds light on the assumptions done by each method (c.g. Gaussian noise assumption, either implicitly as in Notears or explicitly as in Grandag) and, more importantly, simplifies the development of new tools to improve them. For instance, the ideas proposed to deal with partially datascts and non- non-Gaussian noise are readily applicable to any of the causal discovery methods mentioned
in this section, addressing some of their limitations (Kaiser & Sipos 2021: Loh & Bühhmann 2014; Reisach
et al. 2021
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D Datasets Details
Our two benchmark datascts are constructed following similar procodures described in Louizos et al. (2117)
IHDP Hilll, 2011). This dataset contains measurements of both infants (birth weight, head fircumference etc.) and their mother (smoked cigarettes, drank alcohol, took drugs, etc) during real-life data collected in a randomized experiment. The main task is to estimate the effect of home visits by specialists on future cognitive test scores of infants. The outcomes of treatments are simulated artificially as in (Hill) 2011); hence the outcomes of both treatments (home visits or not) on cach subjoct are known. Note that for cach subject, our models are only exposed to only one of the treatments; the outcomes of the other potential potential/counttrrhee /counterfactual outcomes are hidden from the mode, and are only used for the purpose of ATE/CATE /CATE evaluation. To make the task more challenging, additional confoundings are manually introduced by removing a subset (non- white mothers) of the treated children population. In this way we can construct the IHDP dataset of 747 individuals with 6 continuous covariates and 19 binary covariates. We use 10 replicates of different simulations based on setting B (log- linear response surfaces) of (Hill 2011), which can downloaded from https://github.. comm/MMaabbmmsttrrranaffett our models, all continuous covariates are normalized.
We use a 70%/30% train-tost split ratio. Before training
TWINS (Almond et al.) 2005). This dataset consists of twin births in the US between 1989-1991. Only twins which with the same sex born weighing less than 2kg are considered. The treatment is defined being born the heavier one in each twins pair, and the outcome is defined as the mortality of each twins in their first year of life. Therefore, by definition for each pair of twins, we can observe the outcomes of both treatments (the lighter twin and heavier twin). However, during training, only one of the treatment is visible to our models, and the other potential outcome is unknown to the model and are only used for evaluation. The raw dataset is downloaded from hhtpss: h //ggthtb.. AMLab- Ccmm//Maaa-Amtterddmm/Caaa CEVAE Following Louizos et al. (2017), we also introduce artificial confounding using the categorical GESTAT10 variable. This is done by assigning ments (factuals) using the conditional probability 4|x,, =, Bern(a x; + wa (z./ 10//100000 0.1))), where t2 is the treatment assignment for subject i, 2 is the corresponding GESTAT10 covariate, x, denotes the other remaining covariates. Both wp and Wiy are randomly generated as wo ~ (0, 0.1 I), wn ~ N55 0.1). All continuous covariates are normalized.
Ground Truth ATE and CATE Estimation for TWINS and IHDP. In both benchmark datasets, since the held-out hypothetical outcomes of counterfactual treatments are already known, the the ground truth ATE can be naively estimated by averaging the difference between the factual and counterfactual outcomes across the entire dataset. The CATE estimation is a bit tricky, since both datasets contains covariates collected from real-world experiments, in which the underlying ground truth causal graph structure is unknown. As result,, exact CATE estiination is generally impossible for continuous conditioning sets. Therefore, when evaluating the CATE estimation performance on TWINS and IHDP, we focus only on discrete variables (binary and categorical) as conditioning set. This allows unbiased estimation of ground truth CATE by simply averaging the treatment effects on subgroups of subjects in the dataset, that have the corresponding discrete value in the conditioning set. We consider only single conditioning variable at a time, and estimate the corresponding CATE for evaluation.
D.1 CSuite
We develop Causal Suite (CSuite), a number of small to medium (2-12 nodes) synthetic datasets generated from hand-crafted Bayesian networks with the intention of testing different capabilitics of cansal discovery and inference methods. All continuous-only datasets take the form of additive noise models.
Each dataset comes with training set of 2000 samples, and between 1 and 2 intervention test sets. Each intervention test set has a treatment variable, treatment value, reference treatment value and effect variable. We estimate the ground truth ATE by drawing 2000 samples from the treated and reference intervened distributions. For the datasets used to evaluate CATE, we generate samples from conditional intervened distributions by using Hamiltonian Monte Carlo. We employ a burn-in of 10k stops and a thinning factor of 5 generate 2000 conditional samples, which we then use to compute our ground truth CATE estimate. We note that because all ground truth causal quantities are estimated from samples, there is a lower bound
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the expected error that can be obtained by methods. When methods obtain error equal or lower we say that they have solved the task.
7a with linear relationship and Gaussian noise. We have X1 N(O, 1) and X₂ 2 + v Zz where Zz N(0, 1) is independent of X1. The obscrvational distribution is symmetrical in X1 X2. The graph is not identifiable. The best. achievable performance this dataset is obtained when there is uniform distribution over edge direction.
lingauss A two node graph (Figure
2
linexp A two node graph (Figure 7 with linear functional relationship, but with exponentially dis- tributed addilive noise. We have X1 1),, and X2 = x x/3
2 (22 - where Ze Exp(1) is
independent of X1. By using non-Gaussian noise, the graph becomes identifiable. However, the inference problem will be more challenging for methods sensitive to outliers, such as those that assume Gaussian noise.
nonlingauss A two node graph (Figure 7a with nonlinear relationship and Gaussian additive noise. We have X1 (0) and X₂ /6exp(-20 xf) aZ2 where Ze N(0, is independent of X1 and a² = - 6 v :) Note Var(X2) = and Cov(X,, = 0. By having linear correlation of zero between X1 and X2, this dataset creates a potential failure mode for causal inference methods that assume lincarity.
nonlin nonlin_simpsonn simpson A synthetic Simpson's paradox, using the graph Figure 7b if the confounding factor X3 is not adjusted for, the relationship between the treatment X1 and effect X2 reverses. The variable Xa correlates strongly with the effect, but must not be used for adjustment. Choosing an incorrect adjustment set when estimating ([Xx)) leads to significantly incorrect ATE estimate. All variables are continuous, with nonlinear structural equations and non-Gaussian additive noise.
cimprod_simpson Another Simpson's 's paradox using the graph Figure 7c This dalaset is similar to non- lin simpson with 2 key differences: 1) the effect variable the result of a product between the confounding variable and the treatment variable. This makes drawing causal inferences require non-linear function esti- mation. Additionally, the ATE is close 0. The conditioning variable for the CATE task descendant of the confounding variable. This dataset probes for methods' capacity to reduce their uncertainty about confounding variables based on values of its child variables.
large_ bockdooc A nine node graph, as shown in Figure 7d This dataset is constructed so that there are many possible choices of backdoor adjustment set. While both minimal and maximal adjustment sets can result in a correct solution, the a minimal adjustment set results in a much lower-dimensional adjustment problem and thus will result lower variance solutions. The conditioning node for the CATE task is child of the root variable. Thus the CATE task probes for methods' capacity to infor the value of an observed confounder from one of its children. All variables are continuous, with nonlinear structural equations and non-Gaussian additive noise.
weak arrows A nine node graph, shown in Figure 7e Unlike the previous dataset, when the true graph is known, large adjustment set must be used. The causal discovery challenge revolves around finding all arrows, which are scaled to be relatively weak, but which have significant predictive power for X, in aggregate. This dataset tests methods' capacity to identify the full adjustment set and adjust for a large number of variables simultaneously.
to to to__tt__t cts A two node (Figure 7a graph with categorical X, and continuous X2 with an additive noise model. We have X1 Cat ( s(x) = log(exp(1r) 1) is the softplus function, and Zz N(O0 1) is independent of X1.
2 1) takes values in {0, 1,2) and X₂ X, + $ (s(Zz) 1) where
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cts to cat A two node (Figure 7a graph with continuous X1 and categorical X2.
We take X1
V- v3) and X2 categorical on (0, 1, with the following conditional probabilities
6
< i
v3 3
13 13 13
x(22/X1=2)
if
23 σ1
3 3
(32)
1
(3131
()
if >
In this problem, we treat X2 as the trealment and X1 as the target, giving a theoretical ATE zero.
mixed simpson Similar to the nonlin simpson dataset, using the graph of Figure 7b but with X3 categorical on three categories, and X1 binary.
large_ bookddor_bbinrry Similar the bagg__aackooor datasot, using the graph of Figure 7d but with Xg binary.
weak cinowr_biiary.. Similar to the weak arrows dataset, using the graph of Figure 7c but with
Xg binary.
mixed confounding A large, mixed lype dataset with 12 variables, shown in Figure 7 In this dataset, X5 are binary, Xe, Xg are categorical on three categories, and other variables are contimuous, We utilise nonlinear structural equations and non- Gan-ias additive noise.
E
Additional Results
E.1
Causal Discovery Results under Gaussian Exogenous Noise
Figure 4 in the main text shows causal discovery results for the case where synthetic data was generated using non-Gaussian noise. In that case it was observed that using DECI together with flexible noise model performed better than DECI with a Gaussian noise modol. Figure shows results for synthetic data generated using Gaussian noise. As expected, in this case using a Gaussian noise model is beneficial, although DECT with a spline noise mode still performs strongly.
E.2 Summary of CSuite Results
Comprehensive results on CSuite ATE and CATE performance are shown in figures g and 10 provide a summary of results here and then go into per-datasct analysis in the following subsection.
We first
We find DECT to perform consistently well in our 2 node datasets. Tt learns uniform posterior over graphs in the non-identifialle setting, it fits non-linear functions well and it is robust to heavy tailed noise when employing the spline noise model. We find linear and non-linear DML inference to also perform acceptably, with the exception of the heavy tailed noise case, where the methods overfit to outliers and thus estimate ATE poorly.
On the larger (4 and 12 node) datasets, when the true graph is available, DECI provides ATE estimates competitive with the well-established non-linear DMT. method. Notably, DECT outperforms backdoor ad- justment methods when the number of possible adjustment sets large. Choosing the optimal adjustment set is an np-hard problem and the most commin approach is simply choose the largest one. This leads to do dhy suffering from variance. DECI's Limulation-basee approach avoids having to choose adjustment set. On the other hand, for densely connected graphs where the strength of the connection between nodes is low, DECI struggles to capture the funcitonal relationships in the data and DML is most competitive. For CATE estimation DECT provides superior performance in all datasets and is able to completely solve all tasks but one.
When the graph is learnt from the data, the non-linear nature of our (4 and 12 node) datasets together with their heavy tailed noise make the discovery problem very challenging. We find that the PC algorithm
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X1
X2
X2
X2
X1
X2
X3
XA
X3
X1
(a) Two node
(b) Nonlin. Simpson
(c) Symprod. Simpson
X1
X₁
X
X
X₂
X
X4
X5
X
X
X
X7
X
X7
X8
X9
Xs
X
(d) Large backdoor. Treatment Xs, outcome Xs, condition (e)
Weak arrows. X7.
Treatment Xs, outcome Xo, condition
X7.
Xa
X4
X5
X6
Xs
X1
X₂
X7
X9
X10
X12
X11
(f) Mixed confounding
Figure 7: CSuite graphs. Unless otherwise stated, we take X, as the treatment, X2 as the outcome, and for CATE we take X3 as the conditioning variable.
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PC
Notears
DECI-S(PO)
DECI-G(PO)
Notears-Sob
ICALINGAM
DECI-S
DECI-G
Notears-MLP
Grandag
0.8
0.7
0.5
0.4
0.2
0.1
0.8
0.7
Orientation
0.8
0.5
0.2
0.1
Causal Accuracy
0.8
0.6
0.0
ER(16. 16)
ER(16, 64)
ER(64, 64)
ER(B4, 256)
SF(16, 16)
SF(16, 64)
SF(64, 64)
SF(64, 256)
SynTReN
Protein
Figure 8: DECI achieves better results than the baselines in all metrics shown. The plots show the results for causal discovery for synthetic data generated using Gaussian noise. The legend "DECI-G" and "DECI-S" correspond to DECT using a Gaussian and spline noise model. Additionally, the "(PO)" corresponds to running DEC1 with 30% of the training data missing completely at random. For readability, we highlight the DECI results by connecting them with soft lines. The figure shows mean results across five different random seeds.
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CSuite lingauss
CSulte linexp
CSuite nonlingauss
CSulte nonlin simpson
DGs+N
05p
05p+1-
PC+N
T-D5a
T405p
Tel-
0.2
0.4
1.0
CSuite yymprod_simseen simpsen
CSuite large backdoor
CSuite weak_arrows arrows
CSuite cat to cts
D6
05p
05p+1
05p+N
PC1N
14D5a
140%p
T+L-
0.25
1.00
0.2
RMSE
ATE AMSE
CSuite cts to cat
CSuite mixed simpson
CSuite large backdoor binary
CSuite weak arrows arrowssbbiaryy..
DG
05p
05p+1
PC1N
T-DGa
1+0
10
TeL-
0.02
0.04
1.0
0.2 ATE AMSE
0.2
CSult mixed, confounding
DG
050+1
05p
DECI Caussian De Why Linear (DGa
(DGa)
PC DoVVhy Nonlinwar (PC
DoWhy Unear (PCH1
DECI Caussian DeWhh Nonlinear (DGs-N)
True graph DEC Gaussian (T+ (G+)
RC+N
DEC (T405p)
T-DGa
DECI Spiine DoWhty Linear (DSp+L)
True graph DoWhy
T+05p
T+l
DECI Spline DoWhy Nonlieear (D5p++))
Thue graph DoWhy Nonlinear (T+N)
0.0
0.2
1.0
Figure 9: End-to-end ATE results on CSuite.
CSuite nonlin, simpson
CSuite symprod simpson
CSuite large backdoor
DGa
DGa+L
DGa+N
DSp+L
DSp+N
PC+L
PC+N
T+DGa
T+DSp
2
3
0.5
1.0
1.5
2.0
0.00
0.25
0.50
0.75
1.00
RMSE
CATE
Figure 10: End-to-end CATE results on CSuite. Colours and acronyms as in Figure
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Method Dataset
DGa DG DG DSp DSp++L DSp+NN PC+L PC+N
T+L T+N T+DGa D+PSp
ER(16, 16)- G
1.280
1.141
1.129
1.648
1.374
1.393
1.491
1.473
0.945
0.936
1.083
1.332
ER(16, 16)
1.726
1.776
1.780
1.829
1.776
1.755
1.869
1.790
1.755 1.793
1.643
1.660
ER(16, 64)-
1.699
1.422
1,334
1.501
1.442
1,202
1.335
1,440
1.310 1.369
1.460
1.644
ER(16, 64)
2.311
2.465
2.600
2.174
2.452
2,421
2.584
2.276
2.742 2.641
2.510
2.428
ER(61, 64) GG
1.208
1,420
1.190
1.287
1.450
1.397
1.325
1.273
1.284 1.250
1.158
1.124
ER(61, 64)-S
2.246
1.626
1.626 1.892
2.325
2.292
1.526
2.446
2,442 2.411
2.490
2.481
SF(16, 16)
1.156
1.030
1,409
1.699
2.052
1.343
1.574
1.233
1.131 1.078
1.127
1.375
SF(16, 16)
2.870
1.805
2.284
2.431
2.363
1.805
3.008
2.520
2.518 2.502
2.424
2.477
SF(16, 64) G
1.702
1.539
1.635
1.464
1.551 1.510 1.463
1.559
1.486
SF(16, 64)
3.594
4.139
3.877
4.145 4.162 4.106
4.161
3.861
SF(64, 64) GG
1.049
0.998
1.134
1.010
1.035
1.309
0.972
1.343 1.006
1.087
1.288
SF(64, 64) S
2.754
3.239
3.239
3.242
3.239
3.239
3.227
2.591 2.815
2.883
3.034
esuite_cat tt__tts_s
0.495
0.504
0.504 0.501
0.504
0.504
0.262
0.264 0.020 0.023
0.019
0.036
esuite cts to cat csuite large backdoor
0.015
0.011
0.011 0.016
0.011
0,011
0.039
0.038 0.011 0.011
0.011
0.011
0.213
0.144
0.331 0.031
0.091
0.232
1.690
1.690 0. 0.241
0.167
0.035
csuite large__rr________ backdoor bt
0.028
0.029
0,187 0.029
0.029
0.027
0.039
0.039 0,1 0.070
0.021
0.014
esuite _inexp
1.029
1.031
1.031 0.022
0.001
0.002
0.516
0.517 0.001 0.003
0.073
0.028
munite_lingauss
0.120
0.523
0.024 0.149
0.025
0.024
0.498
0.498 0.025 0.022
0.076
0.085
csuite esuite_mixed mixed confounding
0.477
0.461
0.282 0.254
0.500
0.380
0.250
0.387 0.107 0.057
0.019
0.018
csuite_mii siatte_mixed_ssmee
1.259
0.723
0.723 1.765
1.772
1.771
0.723
0.723 0.017 0.022
0.014
0.013
csuite nonlin simpson
1.995
1.994
1.994 1.997
1.994
1.994
1.994
1.994 0.848 0.597
0.404
0.531
csuite satitt__noninngaue
0.042
1.522
1.532 0.043
1.522
1.532
1.532
1.532 1.522 1.532
0.034
0.034
csuite Bes___smppoo_____i simpson
0.318
0.695
0.487 0.427
0.819
0.160
0.487
0.487 0.819 0.168
0.101
0.242
csuite Lace__eaa__arrrrre arrows
1.097
1.108
1.108 0.189
0.110
0.064
1.108
1.108 0.109 0.015
0.255
0.128
_erw_arroww arroosrrwwsll bt
0.226
0.252
0.252 0.085
0.065
0.029
0.123
0.086 0.228 0.003
0.060
0.034
IIDP Twins
0.187
0.187
0.187 0.090
0.101
0.116
0.187
0.187
0.187 0.187
0.146
0.087
0.030
0.025
0.025 0.022
0.025
0.025
0.068
0.025 0.022 0.042
0.022
0.060
Table 3: Median ATE RMSE data underling our rank table. The median is taken across multiple seeds, with the number of sceds shown Table 4 Standard deviations are also shown Table 5 Missing values indicate that the method exceeded the computational budget this typically occurred for larger graphs
provides very poor results or fails to find any causal DAGs compatible with the data when working with these datasets. We find both DECI to provide more acceptable performance with the DECI-spline variant producing more reliable results. In this learnt graph setting, causal inference performance deteriorates sharply consequence of imperfect causal discovery. However, our findings in terms of relative performance among inference methods stay the same.
E.3
Discussion of Continuous CSuite Results
1. lingauss: When the true graph is available, all our causal inference methods are able to solve this problem. However, when the graph needs to be identified from the data, causal discovery accuracy is around 50%. DECI discovery converges to posterior with half of its mass on the right distribution resulting in DECI inference methods showing the lowest error.
2. linexp: The non-Gaussian noise causes difficulties for DECT-Gaussian, which identifies the wrong orientation in majority of cases. As a result, inference algorithm yield poor results. Surprisingly, the PC algorithm is also unable to identify the causal graph, Icading to overall poor inference performance. With the spline noise model, DECI successfully identifies the causal graph, allowing for all inference algorithms to solve the problem.
The non-linear relationship between variables leads all DECT discovery runs to suc- cessfully recover the edge direction for this dataset while PC consistently identifies the wrong edge direction. As expected, linear ATE estimation performs poorly on this task. However, find
nonlingauss:
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Method
DGa DGa+L D DSp DSp+I. DSp+N PC+L PC+N T+L T+N T+DGa T+DSp
Dataset
ER(16, 16)
5
5
5
5
5
5
5
5
ER(16, 16)
5
5
5
5
5
5
5
5
5
ER(16, 61)-
5
5
5
5
5
5
ER(16, 61)- S
5
5
5
3
5
5
5
5
5
ER(64, 64)
5
1
5
4
5
5
5
ER(64, 64)
5
1
5
2
2
1
5
5
5
SF 16)
5
2
5
4
5
5
5
SF(16, 16)
5
1
2
5
2
1
3
5
5
5
5
SF(16, 64)
0
5
4
SF(16, 64)
5
0
O
5
0
O
4
5
5
5
SF(64, 64)
4
5
3
SF(64, 64) S
5
5
5
5
5
3
4
5
5
5
esuite cat tt_oo__oo cts
20
20
20
20
20
20
20
20
20
esuite_ cts to cat
20
20
20
20
20
20
20
20
20
20
20
20
esuite large backdoor
20
20
20
20
20
20
20
20
20
esuite large backdoor bt
20
20
20
20
20
20
20
20
20
20
20
20
esuite linexp
20
20
20
20
20
20
20
20
esuite lingauss
20
20
20
20
20
20
20
20
20
20
20
20
csuite mixed confounding
20
20
20
20
20
20
20
20
20
esuite mixed simpson
20
20
20
20
20
20
20
20
20
20
20
20
csuite nonlin simpson
20
20
20
20
20
20
20
20
csuite_ Soaite_nonlinganss
20
20
20
20
20
20
20
20
20
20
20
20
csuite csuite_symprod harr___ympodd__ppii_peeee simpson
20
20
20
20
20
20
20
20
20
csuite_ weak arrows
20
20
20
20
20
20
20
20
20
20
20
20
esuite weak arrows
20
20
20
20
20
20
20
20
20
IHDP
5
5
5
5
5
5
5
5
5
Twins
5
5
5
5
5
5
5
Table 4: Number of seeds when computing values in Table For ER/SFF graphs, where fewer than 5 seeds were used, this indicates that some rums exceeded the computational budget.
Method
DGa DGa+L DGa+N DSp DSp+L DSp+N PC+L PC+N T+L T+N T+DGa T+DSp
Dataset
esuite caatee__tt_t___ _
0.107
0.210
0.108 0.105
0.194
0.172
0.000
0.000 0.011 0.011
0.000
0.000
esuite__tse _tt___t______ _t__tt___tt___ cat
0.007
0.000
0.000 0.007
0.000
0.000
0.000
0.000
0.009 0.009
0.000
0.000
esuite large__baccddeeeeeeeeee
backdoor
0.724
0.737
0.724 0.041
0.062
0.022
0.000
0.000 0.046 0.028
0.055
0.031
csuite large_ backdoor bt
0.134
0.125
0.124 0.081
0.077
0.056
0.033
0.006 0.011 0.010
0.037
0.004
esuite linexp
0.013
0.000
0.000 0.014
0.000
0.000
0.000
0.000 0.015 0.015
0.000
0.000
esuite lingauss
0.435
0.498
0.432 0.454
0.355
0.489
0.000
0.001 0.020 0.022
0.000
0.000
esuite mixed confounding
0.371
0.389
0.353 0.369
0.418
0.422
0.000
0.000 0.010 0.010
0.000
0.000
csuite_ _iie___iie____ppoonn
0.524
0.522
0.507 0.585
0.522
0.514
0.000
0.000 0.010 0.008
0.000
0.000
Sirite_nonlin____ tti___noonn__sissoonnn
1.071
0.593
0.976 1.080
1.326
1.409
0.000
0.000 0.283 0.304
0.000
0.000
csuite coatte__noinnausss
0.014
0.000
0.000 0.016
0.000
0.000
0.000
0.000 0.013 0.016
0.000
0.000
csuite_syyne _as___smpood_____eeee simpson
0.187
0.162
0.162 0.180
0.000
0.128
0.000
0.000 0.090 0.124
0.000
0.000
esuite weak arrows
0.433
0.401
0.426 0.256
0.118
0.075
0.000
0.000 0.160 0.131
0.000
0.008
esuite _euite_weakkaaeee arrows bt
0.104
0.118
0.113 0.093
0.067
0.067
0.000
0.002 0.020 0.017
0.000
0.001
IHDP
0.021
0.000
0.062 0.013
0.048
0.037
0.024
0.024 0.014 0.022
0.000
0.000
Twins
0.018
0.000
0.000 0.004
0.022
0.019
0.023
0.024 0.008 0.009
0.000
0.000
Table 5: Standard deviations for ATE RMSE results.
DoWhy non-linear not fare much better, likely this is because DML still assumes linear rela- tionship between treatment and target. DECT solves the task successfully.
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Table 6: Method rank on different (CSuite, Twins, IHDP) datasets, ranking by median ATE RMSE. We present the median of the rank over 15 datasets. Supporting data in Table [ We treat methods with access the true graph separately. Note that methods without access to the true graph would typically have a minimum possible rank of 5.
Method
Mean rank
DECI Gaussian (DGa)
DECI Gaussian DoWhy Linear (DGa+L)
9.5
DECI Gaussian DoWhy Nonlinear (DGa+N)
9.5
DECI Spline (DSp)
6
DECI Spline DoWhy Linear (DSp+L)
DECI Spline DoWhy Nonlinear (DSp+N)
4.5
PC + DoWhy Linear (PC+L)
PC +
DoWhy Nonlinear (PC+N)
True graph DECT Gaussian (T+DGa)
True graph DECI Spline (T+DSp)
3
True graph Do Dohhy Linear (T+L)
True graph DoWhy Nonlinear (T+N)
4
nonlin simpson: Even when the true graph is available, none of our inference methods are able recover the true ATE on this more difficult task. We observe non-linear methods (DECI and WoWhy- nonlinear) to perform similarly to each other and more strongly than the simple linear adjustment. For the CATE task, the true value is close to O. This is correctly identified by both and DECT-Spline. Interestingly, we find both linear and non-linear DoWhy variants to overestimate the causal effect when using the backdoor criterion. We attribute this to DECI solving a lower dimensional problem when estimating CATE. While DECT simply regresses the conditioning variable onto the effect variable. The backdoor adjustment employed by DoWhy requires regression from the joint space of conditioning variables and confounders onto the effect variables. The latter procodure involves estimating the relative strength of confounders and conditioning variables, which is more challenging task.
This dataset provides a challenging causal discovery task. DECI identifies the correct edges with probability 0.9. It capacity to recover the edge orientation is slightly worse 0.65. This imperfect causal discovery leads to poor inference for all methods. (potentially because they get Simpson's 's paradox the wrong way around).
symprod_simpson: Even with access to the true graph, no inference method is able to solve this problem. However, we find non-linear methods to clearly outperform linear adjustment for both CATE and ATE estimation. Among non-lincar methods, performance is similar for ATE estimation, with DECI-Gaussian performing slightly better than DoWhy-nonlinear and DECI-Spline slightly worse. However, when estimating CATE, DECT inference present error twice as low nonlinear DoWhy. Again, we attribute this to the backdoor adjustment employed by DoWhy being a more challenging inference task than the 1d regression on simulated data employed by DECI.
In terms of causal discovery, results are similar to nonlin-simpson with PC failing completely and DECI obtaining an adjacency score of 0.92 and orientation of 0.7. The imperfect graph knowl- edge hurts causal inference. Again we see the non-linear backdoor adjustment to perform similarly to DECI for ATE estimation while DECI shows decisively stronger performance when estimating CATE. As expected, the lincar adjustment method fares poorly in this strongly non-lincar setting.
6. weak arrows: When the true causal DAG is available we find that both DoWhy methods solve this ATE problem while both DECI methods predict slightly suboptimal ATE values.
In terms of cansal discovery, DECI elearly outperforms PC with the spline noise model again proving more reliable and leading to better ATE estimates. Although no methods are able to solve the task, find that non-linear DoWhy with the DECI-spline graphs performs best. We hypothesize that
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the amortised function structure employed by DECI suffers in very densely connected graphs with weak edges, like is the case here.
large__ackeeee backdoor: With access the true graph, DECI methods outperform both Dowhy variants for both ATE and CATE estimation. DECI-spline performs best and is able to solve both prob- lems. When faced with many confounders, adjustment procedures suffer from large variance. As a result, despite the non-linearity of the functional relationships at play, the simpler linear backdoor adjustment outperforms the non-linear DMT. approach. On the other hand, DECT's simulation based approach is not disadvantaged in this sctting.
Following the trend of the previous datasets, PC performs poorly in terms of causal discovery, biasing downstream inference methods which perform poorly in terms of ATE and CATE estimation. DECI discovery more reliable, an effect most noticeable when using the spline noise models. With the DECI-Spline posterior over graphs, both DECI-spline and linear DoWhy are able solve the ATE problem and DECI-spline is the only method capable of solving the CATE task. For both tasks and noise models DECI outperforms non-linear DoWhy, again showing its invariance to the size of potential adjustment set.
E.4 Discussion on discrete variables
cat cat__o_ to _t__ttttsssssssttttttt_ttttttt cts We can see when the graph is available, T+DGa performs the best indicating the flexibility of DECT's SEM and effectiveness of the training algorithm. However, when the graph is not available, the performance drops for all methods. This indicates that the discovery part of DECI indeed suffers from such model mis-specification. Changing to discrete variables breaks the model identifiability, which may result in multiple graphs to be learned. This can be partially verified in Table 5 with high variance for DECT methods.
2. cts to_ tat The performances drop is not severe cat_tt_tttt_ee tos_ to cts. From Table 5 the variance is much smaller. This indicates that although this breaks the identifiability, DECT still tends to recover a unique graph. As for the detailed analysis on this scenario, we can leave that for future work.
3. large backdoor bt The performance drop is also minor. From Table 5 we see that the variance is moderate compared to the identifiable case Large_backdoon indicating that DECI still tends to find a unique graph. This shows certain robustness of DECT to model mis-specification.
4. teixed_simpene The performance significantly drops compared to the one with True graph. The high variance in Table 5 also suggests that multiple graphs may be found due to model unidentifia- bility.
weak arrows arrows_binary The performance drop is moderate and also the variance is not high, indicating certain robustness to this mis-specification.
mixed confounding The performance drops and the variance is high, showing that DECI cannot recover the unique graph and is sensitive this type of missspeciiination
From the above analysis of data type mis-specification we can see DECI shows certain robustness. This opens the door for future diroctions regarding which type of data mis-spocification DECT is sensitive to and why.
E.5 Synthetic Graph Experiments
We lest. the performance of DECI on ATE estimation with random graphs as described in section 4.1 For each graph, we randomly generate interventional data for up to five random interventions. We chose the effect variable as the last variable in the causal order that has not yet been used for data generation. For each effect variable we chose the intervention by randomly traversing the graph up three edges away from the effect variable.
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ER(16,
ER(16, 16)
ER(16, 64) G
DGa+Imp.
DGa+L
DGa+N
DSp
mmp
D
DSp
PC+L
PC+N
T4 DG
D+DSp
T+N
ER(16, 64) S
ER(64, 64) G
ER(64, 64)
DGa
DGa+Imp.
DGa+L
N
DSp+Imp.
DSp +L
DSp+ N
PC+L
PC+N
D+
D+
T+L
T+N
10
15
20
25
10
20
30
40
20
40
60
Runtime (h)
Runtime (h)
Runtime (h)
SF(16,
SF(16, 16)
SF(16, 64)
DGa
DGa+Imp.
DGa+L
DG++N
DSp
DSp-Imp.
DSp
DSp+N
PC+L
PC+N-
4+
T DGDGa
DSp
SF(16, 64) 5
SF(64, 64) G
SF(64, 64)
DGa
DGa+Imp.
DGa+L
N
DSp+imp.
DSp+L
DSp+N
PC+L
PC+N
T+DSp
T+L
T+N
10
20
10
20
30
40
50
20
40
60
DECI Gaussian (DGa)
DECI Spline Imputation (DSp+Imp.)
True graph DECI Gaus sian (T+DGa)
DECI Gaussian Imputation (DGa+Imp.)
DECI Spline DoWhy Linear (DSp+L)
True graph DECI Spline (T+DSp)
DoWhy (DGa+L)
Spline DoWhy
(DS++N)
graph DeWhy
DECI Gaussian DoWhy Nonlinear (DGa+N)
DoWhy Linear (PC+L)
True graph DoWhy Nonlinear (T+N)
DECI Spline (DSp)
DoWhy Nonlinear (PC+N)
Figure 11: Runtime of End-to-ond ATE estimation methods on synthetic graphs.
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Table shows the performance of the ATE estimation of DECI and all baselines the synthetic graph data. We only show results for methods that have runtime of less than one day. Figure 11 shows the runtimes for the different methods. DECI has consistently the lowest runtime and scales best to larger graphs. While the runtime of DECI stays approximately constant for various graphs, the runtime of the ATE estimation baselines increases with more complex graphs. In general, the methods using the true graph out purperoom the methods that also perform causal discovery. Further, no method strongly outperforms all other methods with DECI being a strong competitor to the already established DML methods. Lastly, we can see that DECI is capable of performing causal discovery, data imputation and ATE estimation in end-to-end fashion without degrading performance.
E.6 Learning in Non-identifiable Settings with the Help of Graph Priors
We investigate the utility of prior knowledge over causal graphs for causal discovery and end2end inference in non-identiffable and difficult to identify settings. Specifically, we generate 2 datasets composed of 2000 training examples each. The first is composed of only linear relationships between variables and Gaussian additive noise, making the causal graph non-identifiable The second dataset also uses linear functions but has a mix of exponential and challenging.
noise. Although identifiable, discovery in this latter setting is
We introduce prior knowledge about graph sparseness through the weighted adjacency matrix Wo E [0, Dx with zero entries encouraging sparser graphs. The resulting informed DECI prior is
p(G) x exp (-X,IIG Woll ph(G)² a h(G).
with the scalar X₂ regulating the strength of the prior beliefs encoded Wo-
We compare DECI inference with access to the true graph to end2end DECI inference. In the latter case consider PC prior, which has as its mean the CP-DAG provided by PC. We consider different prior strengths, i.c. the value of the entries of Wo, between 0 and 1. We also experiment with introducing the true-graph as prior of this form, yielding what we refer to as the "informed prior".
In the noniidentifiable case, we find both DECI (prior strength 0) and PC discovery to provide incorrect graphs. Interestingly, providing the PC CPDAG as a prior for DECI can yield large gains terms of causal discovery due a variance reduction effect. These gains do not translate to better ATE estimation, where performance is not improved over the uninformative prior. Providing knowledge of the true graph does help causal inference, with a more confident prior yielding better results.
In the difficult identifiable case, the PC prior does provide gains to DECL We find the optimal prior strength to be 0.5: a balanced combination of PC and DECT discovery is most reliable, while using exclusively one of the two algorithms yields worse results. In this identifiable setting informed DECI discovery is able to obtain perfect ATE estimation performance with a prior strength as low as 0.2.
E.7 Understanding Robustness to Unobserved Confounders
To evaluate the robustness of DECI to the presence of unobserved confounders, we introduced a new CSuite datasets. Fork nonlin. Gauss has the graph shown in Figure[4aa [4a uses nonlinear functions with Gaussian exogencous noise. We trained DECI on the data from observed variables only (as shown in Figure 14), as well as on the full dataset treating all variables as observed. Table shows the results.
Table 7: ATE RMSE results on CSuite Fork nonlin. Gauss dataset with unobserved confounders.
Method
ATE RMSE (mean + s.e. from seeds)
DECT Gaussian (DGa)
0.034 + 0.007
DECT Spline (DSp)
0.036 + 0.009
DECT Gaussian (DGa) full observability
0.019 + 0.002
DECI Spline (DSp) full observability
0.018 + 0.002
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linear Gaussian non-identifiable
adjacency ffscor
1.0
orientation fscore
1.0
ATE RMSE
0.2
0.9
0.8
0.8
0.1
0.7
informed prior
prior
0.2
graph available
0.0
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.8
1.0
prior strength
prior strength
prior strength
Figure 12: Causal discovery and inference results obtained on 9 node linear Gaussian dataset, where the graph is non-identifiable without prior knowledge. We perform DECI inference with the true graph and DECI end2end inference with different priors. Informed prior refers to using smoothed version of the true graph the prior. PC prior refers to using the CP-DAG outputted by PC as the prior mean Wo. The prior strength indicates how much prior mass is placed on the mean prior graph Wo and how much is spread across all other DAGs.
linear semi-Gaussion
0.3
adjacency ffccrr
1.0
orientation [score
1.0
ATE RMSE
0.2
0.8
0.8
Informed prlor
0.1
0.6
prior
0.4
true graph available
0.0
0.6
0.8
1.0
0.2
0.4
0.6
0.0
0.2
0.4
0.8
1.0
prior strength
prior strength
prior strength
Figure 13: Causal discovery and inference results obtained on 9 node linear non- non-Gaussian dataset, difficult to identify without prior knowledge. We perform DECI inference with the true graph and DECI end2end inference with different priors. Informed prior refers to using smoothed version of the true graph as the prior. PC prior refers to using the CP-DAG outputted by PC as the prior mean Wo. The prior strength indicates how much prior mass is placed on the mean prior graph Wo and how much is spread across all other DAGs.
F Additional Related Work
Similarly, recent works have combined the continuous DAG learning approach with variational inference Lorch al.] 2021 Annadani et al. 2021 Cundy et al. 2021 Deleu et al. 2022). However, most of them locus linear Gaussian models (Annadani et al. 2021 Cundy et al. 2021 Deleu et al. 2022). Only Lorch et al. (2021 considers non-linear functional forms but is constrained to Gaussian noise models. To the best of our knowledge, we are the first to present an identifiability proof for causal discovery using VI.
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X₂
X1
X3
(a) Fork.
Treatment X2, outcome Xg.
Figure 14: CSuite graphs with unobserved confounder.
42


Reviews:
[{"review_id": "FKoZ4RpY1C", "reviewer": "Reviewer_RTJh", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strength:\n1. I like the proposed framework which works for both the causal discovery and causal inference, where the causal discovery is via learning the variational distribution $q_{\\phi}(G)$ by ENCO parameterisation, and the causal inference is estimated by constructing MLPs for each node with function $f_i(\\mathbf{x})$ for characterizing $p_{\\theta}(\\mathbf{x}^n|G)$. This framework is a unified view of functional causal discovery including many previous literatures such as Notears, GAE and Grandag. \n2. Further, the method is supported by theoretical analysis ad could be extended to more real scenarios such as mixed-type data and missing data.\n3. This paper is excellently written, with a clear structure and methodology. The explanations across various aspects are commendable, and the overall framework is convincing. It is well-supported by overwhelming experiments and theoretical evidence, making for a very enjoyable reading experience.\n\n## Weakness:\nNevertheless, while I really like the framework the paper proposed, I do have a few questions as list below.\n### Major Comments:\n1. Regarding the interpretation of the parameter $\\theta$. There are two difference concepts here, one is the true parameter $\\theta^*$ for the true function $f_i$ (if we assume it is parametric), and another is the estimated $\\tilde{\\theta}$ that is characterised by $\\ell_i$ and $\\zeta_i$. It seems the paper treats them as the same concept throughout the paper where the flexible parameterisation is matching with the true parameterisation. Also the same confusion rises up on the correct specification where $p_{\\theta^*}$ would have the same structure as the MLPs and through maximising the ELBO. Then the MLP parameters will be converging to the true parameters with a large number of data. However, in most real cases it may not be true, do the authors have more comments on misspecified models where the functional estimation varies from the true estimates?\n2. While I really appreciate the great efforts that author made on the performance evaluation, it would be interesting to see the discussions on the training sample sizes as they are fixed across some of the experiments. It would be interesting to add a few discussions on the effectiveness of the framework with different training sample sizes. \n3. The framework consists of many surrogate components as listed below. It would be helpful to find out which components would lead to the most effective estimation for getting more insights. Also it would be good to know the training and inference complexity for different tasks. \n    - Prior over $G$ \n    - Parameterization of $f_i$\n    - Parameterization of $q_{\\phi}$\n    - $g_G$ with different basis-function linear models\n    - Hyper-parameters of training of MLPs\n\n\n\n### Minor Comments:\n1. It seems that $f_i$ is assumed to be parametric at the first place. Any comments on the scenarios where $f_i$ is non-parameteric?\n2. “A naïve implementation would require training 2D MLPs. Instead, we construct these MLPs so that their weights are shared across nodes…” This sentence is a bit confusing, do you mean instead of training 2D MLPs, we only train 2 MLPs for $\\ell$ and $\\zeta$, respectively across different nodes?\n3. Abuse of notation where $g_G$ on (13) is the same as (7).\n4. Some notations could be more clearly specified such as Hadamard product on (5), Fronbiues Norm on (6), and dirichlet function $\\delta$. \n5. It would be great to give more intuitions on the parameterization of ENCO in the context for $q_\\phi$.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a unified framework that performs both causal discovery and causal inference simultaneously from observational data, namely deep end-to-end causal inference (DECI). This work  contributes to the field of causal inference by addressing a longstanding challenge that combines the causal discovery and causal inference processes within the same framework by the Bayesian graph structure. Key contributions of the paper include:\n\n1. Introduction of an end-to-end non-linear additive noise model that utilizes neural network functional relationships for both causal discovery and inference, including the estimation of average treatment effects (ATE) and conditional ATE，.\n2. Theoretical guarantees for the framework are provided, demonstrating that DECI can asymptotically recover the true causal graph and treatment effects, assuming the model is correctly specified with the true parameters.\n3. DECI's performance is empirically validated through extensive experiments, showing its competitive or superior performance compared to existing methods in both causal discovery and casual effect estimation across a variety of synthetic, real and benchmark datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weakness above.", "broader_impact_concerns": "I don't have any concerns about the broader impact of this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bggBBmgdvy", "reviewer": "Reviewer_Xkxj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strength \nThe work effectively demonstrates the practical utility of its framework through a range of examples, from synthetic to real-world scenarios. Its most notable strength lies in providing a comprehensive end-to-end solution for causal inference, encompassing both causal discovery and estimation processes.\n\n\n# Weakness\nThe overall soundness of the proposed framework is questionable, primarily due to its reliance on a series of approximations and assumptions. This raises doubts about the reliability of the resulting estimated Average Treatment Effect (ATE) / Conditional Average Treatment Effect (CATE). Compared to Markov-equivalence-class (MEC)-based methods like those of Jung et al., 2021, and Guo et al., 2021, this framework is predicated on more stringent assumptions, such as the absence of unmeasured confounders, third-order differentiability of functions, specific noise structures, and correctly specified models. In contrast, the existing methods utilizing the Partial Ancestral Graph (PAG) or Completed Partially Directed Acyclic Graph (CPDAG) only rely on the assumption of a correct Conditional Independence (CI)-oracle.\n\nFurthermore, the reliance on the NOTEARS method, an approximate approach for discovering the true causal graph, adds another layer of uncertainty. The method is possible to select a graph outside the Markov equivalence class, which may lead to a misspecified graph. Given that this approximation influences both the learning of the graph and the estimation of causal effects, the resultant causal effect estimate is not reliable.\n\nFinally, the paper falls short in adequately comparing its approach with previous works. For instance, Jung et al., (2021b) and Guo et al., (2021) also offer end-to-end causal inference frameworks, leveraging the PAG, a summary of the MEC allowing for unmeasured confounders. These methods, utilizing readily available causal discovery techniques, warrant a more transparent acknowledgment and detailed comparative analysis.\n\n# Minor issues \n\n1. The articulation of the contributions could be more accessible to readers. The paper should specifically elaborate on the nature of the graph posterior in the contribution section.\n2. The absence of theoretical guarantees regarding the convergence of the estimated causal effect to the true causal effect, including the convergence rate, is a significant omission.\n3. Clarity is needed on what novel contributions this method provides, especially in comparison to existing methods like those of Jung et al., (2021b) and Guo et al., (2021). What distinguishes this method from others in the field?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This study introduces a deep-learning-based framework that spanning from causal discovery to causal effect estimation. Operating under specific structural assumptions, including the absence of unmeasured confounders, and making parametric assumptions about noise, it employs the NOTEARS-based approach for causal discovery. Following this, the framework uses the posterior distribution over the graph, denoted as q_{\\phi}(G), to derive the interventional Structural Causal Model (SCM). This facilitates the estimation of causal effects from these models. The authors highlight the practical advantages of their proposed model through a variety of examples, underlining its applicability and effectiveness.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address the concerns described in the Weakness and Minor issues.", "broader_impact_concerns": "This work doesn't possess any ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DziRKeBcqJ", "reviewer": "Reviewer_DoeS", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I have read some previous versions of this paper and seen the improvements made by the authors. This paper contributes to integrating the causal structure learning task and the downstream causal inference task into a unified learning framework, which is quite meaningful from my perspective. The paper is very well written and polished. The topic is of theoretical and practical importance. Moreover, the authors have provided really extensive experimental results to show the effectiveness of the proposed method, making this paper quite solid.\n\nHowever, there are a few weaknesses in the paper that could be addressed:\n\n1. The authors should clarify why using the learned graph makes treatment effect estimation more efficient. While prior knowledge of treatment and outcome can be used to estimate Conditional Average Treatment Effects (ATE) and ATE directly using methods like Slearner, Tlearner, and Xlearner, it would be beneficial to explain how the learned graph improves efficiency in this context.\n\n2. Equation 9 restricts the noise distributions to be a fixed Gaussian, which may not accurately reflect real-world scenarios where noise distributions are not equal-variance Gaussian. It would be helpful if the paper discussed how this restriction affects the learning results of the proposed method.\n\n3. The method's effectiveness in causal discovery may be limited when the dimensionality of the variables is high (more than 100), which could impact the results of causal inference. It would be informative to see the results of this scenario and compare the method with well-known Targeted Maximum Likelihood Estimation (TMLE) methods, such as TarNet.\n\n4. In Figure 3, the probabilistic property allows the method to incorporate the results of multiple learned causal graphs to estimate treatment effects. However, if one of the learned graphs is incorrect but has a lower probability, it would be helpful to discuss how this would affect the inference results.\n\nAdditionally, it would be beneficial to elaborate more on the similarities between the proposed method and \"Deep Structural Causal Models for Tractable Counterfactual Inference\" to provide a clearer understanding of the contributions of the proposed method to existing literature.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces the Deep End-to-end Causal Inference (DECI) framework, proposed to address challenges in causal inference. The proposed framework is a flow-based non-linear additive noise model combined with variational inference, enabling Bayesian causal discovery and inference. The framework unifies many existing structural equation model (SEM) based causal inference techniques and can recover the ground truth mechanism under standard assumptions. Additionally, DECI can be extended to handle heterogeneous, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Extensive empirical evaluations have been conducted to demonstrate DECI's competitive performance compared to relevant baselines for both causal discovery and inference.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weaknesses.", "broader_impact_concerns": "No concerns about broader impacts.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: A majority of the reviewers supported accepting the paper but pointed out small revisions that could improve it. In particular, for the camera-ready version, reviewers asked that the following be included: 

* A comment on whether causal effects are always identifiable in the learned graphs, and if not, what the algorithm does in the case where this fails. 
* A comment on the sample efficiency of the method in relation to the literature, such as Jung et al., (2021b).
* An elaboration (in the main paper) on the difference between $\theta$ and $\theta^*$, potential model misspecification, and misestimation, both in the case where $\theta^*$ is unique and where it is not. Currently, there is only a brief comment in the appendix.
Decision: Accept (poster)