Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries
Authors: Patrick Chao
Patrick Blöbaum
Sapan Kirit Patel
Shiva Kasiviswanathan
Venue: Accepted by TMLR ()

Abstract: We consider the problem of answering observational, interventional, and counterfactual queries in a causally sufficient setting where only observational data and the causal graph are available. Utilizing the recent developments in diffusion models, we introduce diffusion-based causal models (DCM) to learn causal mechanisms, that generate unique latent encodings.  These encodings enable us to directly sample under interventions and perform abduction for counterfactuals. Diffusion models are a natural fit here, since they can encode each node to a latent representation that acts as a proxy for exogenous noise. Our empirical evaluations demonstrate significant improvements over existing state-of-the-art methods for answering causal queries. Furthermore, we provide theoretical results that offer a methodology for analyzing counterfactual estimation in general encoder-decoder models, which could be useful in settings beyond our proposed approach.

Published in Transactions on Machine Learning Research (11/2024)
Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries
Patrick Chao University of Pennsylvania
pchao @ www.ton.upann.cc.....
Patrick Blöbaum Amazon
bloebp ammzoon.cammmmm
Sapan Patel
sapanp@amazon.com
Amazon
Shiva Prasad Kasiviswanathan Amazon
kasivisw gmail.com
Reviewed on http:////w.ooooo///////////////a
Abstract
We consider the problem of answering observational, interventional. and counterfactual queries in causally sufficient setting where only observational data and the causal graph are available. Utilizing the recent developments in diffusion models, we introduce diffusion-based causal models (DCM) to learn causal mechanisms, that generate unique latent encodings. These encodings enable us to directly sample under interventions and perform abduction for counterfactuals. Diffusion models are a natural fit here, since they can encode each node to a latent representation that acts proxy for exogenous noise. Our empirical evaluations demonstrate significant improvements over existing sfate-of-the-ar methods for answering causal queries. Furthermore, we provide theoretical results that offer methodology for analyzing counterfactual estimation in general encoder-decoder models, which could useful in settings beyond our proposed approach.
Introduction
Understanding the causal relationships in complex problems crucial for making analyses, conclusions, and gen- eralized predictions. To achieve this, we require causal models and queries. Structural Causal Models (SCMs) are generative models describing the causal relationships between variables, allowing for observational, interventional, and counterfactual queries (Pearl 2009a). An SCM specifies how set of endogenous (observed) random variables is generated from a set of exogenous (unobserved) random variables with prior distribution via a set of structural equations.
Given a causal DAG. we focus on approximating the individual SCMs with diffusion models. As an application, we present flexible framework for answering all the three types of causal (observational, interventional, and counterfac- tual) queries. For counterfactuals, we work in Pearl's SCM framework (Pearl) 2009a), and seek to quantify unit-level statements of the form: Given that observed a factual sample (......, x for set of K variables (X)....,X, what would have been the outcome for these K variables be, if the value of some set Xx (with C [K]) had been set some y E ? Throughout this paper we assume causal sufficiency, i.c., absence of hidden confounders. Note that causal sufficiency is necessary assumption for answering causal queries from observational data alone.
In the SCM framework, causal queries can be answered by learning a proxy for the unobserved exogenous noise and the structural equations. This suggests that (conditional) generative models that encode to a latent space could be an attractive choice for the modeling SCMs, the latent serves as the proxy for the exogenous noise. In these models,
Work done during internship Amazon.

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the encoding process extracts the latent from an observation, and the decoding process generates the sample from the latent, approximating the structural equations.
Our Contributions. In this work, we propose and analyze the effectiveness of using a diffusion model for modeling
SCMs. Diffusion models (Sohl-Dickstein al. 2013 Ho et al.) 2020 Song et al. [021 have gained popularity recently due to their high expressivity and exceptional performance in generative tasks (Saharia et al. [022] [aamehh
et al. 2022, Kong et al. 2021). The primary contribution of our work is in determining how to apply diffusion models to the causal setting. Rather than focusing a single causal inference setting, our aim through this modeling is to provide a single flexible framework that works for answering a wide range of causal queries. Now applying diffusion models to the causal setting is non-trivial because diffusion models are typically used to learn a stochastic process mapping between standard Gaussian and data distribution. Our main idea is to model each node in the causal graph diffusion model and cascade generated samples in topological order to answer causal queries. For each node, the corresponding diffusion model takes as input the node and parent values to encode and decode a latent representation. To implement the diffusion model, we utilize the recently proposed Denoising Diffusiio Implicit Models (DDIMs) (Song al., 2021), which may be interpreted dimensionality reduction while encoding, We leverage the deterministic forward and backward diffusion processes of DDIMs to use diffusion models as an encoder-decoder model. We refer to the resulting model as diffasion-based causal model (DCM) and show that this model mimics the necessary properties of an SCM. Our key contributions include:
deterministic autoencoder model without any
(1) [Section]] We propose diffusion-based causal model (DCM), new model class for modeling structural causal models, that provides a flexible and practical framework for approximating both interventions (do-operator) and counterfactuals (abduction-action)) aetain--peeeicttttee
steps). We present a procedure for training a DCM given just the causal graph and observational data, and show that the resulting trained model enables sampling from the observa- tional and interventional distributions, and facilitates answering counterfactual queries.
(2) [Section Our theoretical analysis examines the accuracy of counterfactual estimates generated by the DCM, and we demonstrate that they can be bounded given some reasonable assumptions. Importantly, our analysis is not limited to diffusion models, but also applics to other encoder-decoder settings. To the best of our knowledge, these are the first error bounds that explain the observed performance improvements in using encoder-decoder models, like diffusion models, to address counterfactual queries. Another feature of this result, is that it also extends, under an additional assumption, to the more challenging multivariate case.
(3) [Section 5) We evaluate the performance of DCM on a range of synthetic datasets generated with various structural equation types for all three forms of causal queries. We find that DCM consistently outperforms existing state- th-the-art methods (Sánchez-Martin et al./ 2022 Khemakhem al. 2021 In fact, for certain interventional and counterfactual queries such as those arising with nonadditive noise models, DCM is better by order of magnitude or more than these existing approaches. Additionally, we demonstrate the favorable performance of DCM on an interventional query experiment conducted on fMRI data.
Related Work. Over the years, variety of methods have been developed in the causal inference literature for answer-
ing interventional and/or counterfactual queries including non- con-paraettric methods (Shalit et al., 2017: Alaa & Van
Der Schaar 2017] Muandet al.] (021) and probabilistic modeling methods (Zečević et al. 2021a) la) More relevant to
our approach is recent series of work, including (Morafiah et al., 2020 Pawlowski al. 2020 Kocaoglu et al. 2018,
Parafita & Vitria 2020 Zečević et al. 2021bb Garrido et al.) 2021 Karimi et al.) 2020 Linchez-Martin et al. 2022
Khemakhem et al. 202 Sanchez & Tsaftaris 2022) that have demonstrated the success of using deep (conditional)
generative models for this task.
Karimi al. (2000) propose approach for answering interventional queries by fitting a conditional variational autoencoder to each conditional in the Markov factorization implied by the causal graph. Also using the ideas of variational inference and normalizing flows, Pawlowski et al. (2020) propose schemes for counterfactual inference.
In Khemakhem et al. 2022), 1), the authors propose an autoregressive normalizing flow for causal discovery and queries, referred to as CAREFL. CAREFL is also applicable even with only knowledge of the causal ordering rather than the full causal graph as we require. However, as also noted by (Sänchez-Martin al. (2022), when the causal graph is present, CAREFL is unable to exploit the absence of edges fully as it reduces a causal graph to its causal ordering (which may not be unique). Using normalizing flows for answering causal queries with causal DAG was also explored by (Balgi let al. 2022). however their approach does not have any theoretical guarantees on their counterfactual estimates, and their experimental evaluation is quite limited.
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Sánchez-Martng al. (2022) propose VACA, which uses graph neural networks (GNNs) in the form of a variational graph autoencoder to sample from the observational, interventional, and counterfactual distribution. VACA can uti- lize the inherent graph structure through the GNN, however, suffers in empirical performance (see Section (5). Fur- thermore, the usage of the GNN leads to undesirable design constraints, e.g., the encoder GNN cannot have hidden layers (Sánchez-Martin et al. 2022).
A very recent work by Javaloy et al. (2023) combines the idea of using autoregressive normalizing flows (as in CAREFL) and modeling the entire causal graph as one model (as in VACA), to reduce possible error propagation in the graph. The authors further generalize the theoretical results from Khemakhem et al. (202I) beyond affine autoregressive normalizing flows and establish a clearer connection to SCMs by providing more direct way of ap- plying the do-operator. In contrast to our work, Javally ct al. (2023) focuses modeling the whole causal graph as one model. Modeling on per-node basis has several advantages over a single model in terms of flexibility and computational efficiency because it permits individual node models to be trained in parallel. For example, on a single experiment, we observed that our DCM approach trains roughly seven times faster than CAREFL, and nine times faster than VACA (see Appendix D.I.
Sanchez& & Tsaftaris 2022) use diffusion models for counterfactual estimation, focusing on the bivariate graph case with an image class causing an image. The authors train a diffusion model to generate images and use the abduction- action-prediction procedure from Pearl et al. (2016) as well as classifier guidance (Dhariwal & Nichol 202T) to generate counterfactual images. However, this is solely for bivariate models and requires training separate classifier for intermediate diffusion images, and exhibits poor performance for more complex images e.g., ImageNet (Deng ct al. 2200). Our approach distinguishes itself from Sanchez& & Tsaftaris (202)) as it can handle general causal graphs (beyond the simpler two node setting) and operates on continuous variables (beyond the simpler case of a discrete label and image). Our experimental evaluations are more general as it also covers interventional queries not addressed by [Sanchez& & Tsaftaris (2022). Finally, in terms of theoretical contribution, we provide rigorous conditions the latent model and structural equations under which we can estimate counterfactuals. Even for two variable setting considered by Sanchez & Tsaftaris (2022) such a theoretical understanding was previously missing.
Finally, a diffusion model based approach has also been recently proposed for the different task of causal discovery under additive noise models, where the diffusion model serves as an approximation to the Hessian functions (Sanchez) et al.) 2022). This raises the interesting question of whether the diffusion models can be adapted for solving the end-to-end problem from discovery to inference.
2 Preliminaries
Notation. To distinguish random variables from their instantiation, we represent the former with capital letters and the latter with the corresponding lowercase letters. To distinguish between the nodes in the causal graph and diffusion random variables, we use subscripts to denote graph nodes. Let [n]] := [I....,.).
Structural Causal Models. Consider a directed acyclic graph (DAG) with nodes {l...., } in a topologically sorted order, where node i is represented by (random) variable X; in some generic space X; C Rdd, Let pa, be the parents of node in G and let X Xps; = {X,} hepe, be the variables of the parents of node i. A structural causal model M describes the relationship between an observed/endogenous node and its causal parents. Formally, an SCM . == (F, p(U)) determines how a set of K endogenous random variables X == (X,,... .x> generated from a set of exogenous random variables U == (UI,..., Ux} with prior distribution p(U) via set of structural equations, F == (J1...., where X, := J((pp,, U.) for i E [K]. Throughout this paper, we assume that the unobserved random variables are jointly independent (Markovian SCM), and the DAG G is the graph induced by M. Every SCM M entails a unique joint observational distribution satisfying the causal Markov assumption: XXX) = -I (X,1 | Xpa,).
Structural causal models address Pearl's causal hierarchy (or "ladder of causation"), which consists of three "layers" of causal queries in increasing complexity (Pearl, 2009a): observational (or associational), interventional, and coun- terfactual. As example, an interventional query can be formulated as "What will be the effect the population X, if a variable X; is assigned a fixed value ?t?" The do-operator do(X; := 7)) represents the effect of setting variable X; to "/-- Note that our proposed framework allows for more general sets of interventions as well, such as interventions on multiple variables denoted as do(X, == 7) (where I S [K]. Xx= == (Xi)ke ? E Rl An inter- vention operation, do(Xz Xx := 7), transforms the original joint distribution into an interventional distribution denoted
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by p(X do(Xz ?)). On the other hand, counterfactual query be framed as "What would have been the outcome of particular factual sample z" := (zf..... .......,,, k, if the value of Xz had been set to y?".. Counterfactual estimation may be performed through the three-step procedure of 1) abduction: estimation of the exogemous noise U, 2) action: intervene do(Xz == 7), and 3) prediction: estimating zCF using the abducted noise and intervention values, Note that in Step we perform deterministic counterfactual reasoning. focusing counterfactuals pertaining single unit of the population.
Diffusion Models. Given data from distribution X ~ Q, the objective of diffusion models is to construct an effi- ciently sampleable distribution approximating Q. Denoising diffusion probabilistic models (DDPMs) (Sohi-Dickstein) et al. 2015 Ho al., 2020) accomplish this by introducing forward noising process that adds isotropic Gaussian noise cach time step and a learned reverse denoising process. A common representation of diffusion models is fixed Markov chain that adds Gaussian noise with variances 31....,87 Br E (0, 1), generating latent variables X ..... .'....."'' X
q(x' x² 11-1) N(' vT 3,x'11 8., J) and q(x* =0) = N(X:; Vara", -axa))), where = II: 1 (1 B.).
Here, T E zt, and & E {o...., T denotes the time index.
By choosing sufficiently large T and ai that converge to O, we have X is distributed as an isotropic Gaussian distribution. The learned reverse diffusion process attempts approximate the intractable q ( x²) using neural network and is defined as a Markov chain with Gaussian transitions, po x²) WXX ; pa (z t), S (z' .t)). Rather than predicting
directly, the network could instead predict the Gaussian noise & from x' fazt +
VT- oye. Ho et al. (2020) found that modeling e instead of μæ, fixing Σg, and using the following reweighted loss function
Uni [||= se(Va + VI a,e, t)] x=~Q
(1)
e~N((11)
works well empirically. We also utilize this loss function in our training.
Song al. (202I) demonstrate that it possible to take a pretrained standard denoising diffusion probabilistic model (DDPM) and generalize the generation to non-Markovian processes. In particular, it is possible to pretrained
DDPM model to obtain a deterministic sample given noise X", known as the denoising diffusion implicit model (DDIM), with reverse implicit diffusion process
X
a ai-1xt Ex(X (Vai-1(1-au)) - / a6-1
at
Note that the X here is deterministic. We also use a forward implicit diffusion process introduced by Song et al. (202T), derived from rewriting the DDIM process Eq. [ as an ordinary differential equation (ODE) and considering the Fuler method approximation in the forward direction to obtain
X++1
act1 x (X( (VI- 0+41 Voeti(1 10111-a4//aa ax)/oz an
We utilize the DDIM framework in this work, in particular Eqs. and 2 will define the encoding (forward) and decoding (reverse) processes. Note that this ensures deterministic encoding and decoding. This construction produces unique latent variable per observation, as well as a unique decoding, and also ensures we obtain the same output for repeated counterfactual queries.
3 DCMs: Diffusion-based Causal Models
In this section, we present our DCM approach for modeling the SCMs and to answer causal queries. The DCM approach falls in a general class of techniques that try to model a structural causal model by using an encoder-decoder pair. Consider a data generating process X f ((Xa..)) The goal will construct an encoding function g and decoding function h. The encoding function g attempts to represent the information in U: for pair (X, pa) Z = g(X, Xpa) is the latent variable. The decoder takes the input Z and Xpa as input to attempt to reconstruct X: x h(Z, Xp where under perfect reconstruction, X X. The decoding function mimics the true structural
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equation f, although it does not need to be exactly equal. For example, there are infinitely many encodings that satisfy Z = r()) for all U for an invertible function r.
We first explain the construction and the training process of DCM, and then explain how the model can be used for answering various causal queries. We start with some notations.
Define Z to be the ith endogenous node value at diffusion step of the forward implicit diffusion process (Eq. 3). and let Zi == Z
Define X to be ith endogenous node value at diffusion step & of the reverse implicit diffusion process (Eq, 2 and let X; = X
Training DCM. We train diffusion model for each node, taking denoised parent values as input. The parent values can be interpreted additional covariates to the model, where one may choose to use classifier free guidance to incorporate the covariates (Ho & Salimans 2021 Empirically, we find that simply concatenating the covariates results in better performance than classifier free guidance.
We use the Eo parametrization for the diffusion model from Ho al. (2020), representing the diffusion model for node as (x, pp,, t)- The complete training procedure presented in Algorithm is only slightly modified from the usual training procedure, with the additions of the parents as covariates and training a diffusion model for each node, Since the generative models learned for generation of different endogenous nodes do not affect training of each other, these models may be trained in parallel. For each node in the graph, we can train a model in parallel as each diffusion model only requires the current node and parent values. Our final DCM model just combination of these K trained diffusion models Eg.. Eg......
Algorithm DCM Training
Input: Distribution Q, scale factors {ar) causal DAG with node represented by X;
1: while not converged do
2: Sample X ~ Q
3: for i 1,..., K do
t~ Unif((...... .TH
E N(0, Update parameters of node i's diffusion model Eq. by |e X/ -1- x pa, 1)
(d. is the dimension of X,) minimizing the following loss: (based on Eq.
7: end for 8: end while
We use all the variables (X1...., .xk for the training procedure, because a priori, we do not assume anything on the possible causal queries, i... we allow for all possible target variables, intervened variables, etc. However, if we are only interested in some pre-defined set of queries, then the graph could be reduced accordingly. For example, if we are only interested counterfactual estimate of particular node with respect to an intervention of a predecessor, one can simply reduce to a subgraph containing the target node, the intervened node and a backdoor adjustment set (e.g.. the ancestors of the intervened node). This then reduces to only learning single diffusion model.
One major advantage of our proposed DCM approach is the ability to generalize to larger graphs. Since each diffusion model only uses the parents as input, modeling each node depends only on the incoming degree of the node (the number of causal parents). While the number of diffusion models scales with the number of non-root nodes, each model is generally small in terms of its parameter size and can be trained in parallel. Additionally, we may apply the proposed DCM approach to any setting where diffusion models are applicable: continuous variables, high dimensional settings, categorical data, images, etc.
Encoding and Decoding Steps with DCM. With a DCM, the encoding (resp. decoding) process is identical to the DDIM encoding (resp. decoding) process except we include the parent values as additional covariates. Note that, given the model eg, DDIM is deterministic process laid out in Eqs. [nd Let focus on node E [K] (same process is repeated for each node i). The encoding process takes X; and its parent values X pa, as input and maps it to a latent variable Z, The decoding process takes Z, and Xnn, as input to construct X, (an approximation of X,). Formally, using the forward implicit diffusion process in Eq. given a sample X;, encode unique latent variable
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Zi zí using the recursive formula
7,t1
a++1 7 + cg(Zi pp 1) Ot
Vi-acHi act1
ac+1(1 -a))
v2 = o,.., 7
(4)
where 2 = X4. The latent variable Zj acts proxy for the exogenous noise U,. Using the reverse implicit
diffusion process from DDIM Eq. 2 given latent vector Z₁ obtain deterministic decoding X; == X using the recursive formula
cx xpa: :t) ( (
oz) (oc-111-0))
(0-1(1-00)
x
X
/1 0t-1 for all I,....I,
(5)
at
at
where XT := Zi In the following, we use Enc,(X: X Xpa, and Dec, (Z,, Xpa,) to denote the encoding and decoding functions node defined Eqns. and respectively. See Algorithms and for detailed pseudocodes.
Algorithm Enc; (X,, Xpa.)
Input: X,, Xpu,
1: 2 +- X,
for O,.... T do
Z
aeti Zi + (Z) X :t)
(VI-ou+)
ax+1 a2
(1-1(1-o))
4: end for
5: Return Zi = Z
Algorithm 3 Dec; (Zi, Xpa,)
Input: 7,, Xpa,
1: R t- Zi
2: for T T..... do
(va az
8.-1
ac 81
(8,
X
:pa,:)
VI at-1
4: end for
5: Return 8, = 8
Answering Causal Queries with Trained DCM. We now describe how trained DCM model can be used for (approximately) answering causal queries. Answering observational and interventional queries require sampling from the observational and the interventional distribution respectively. With counterfactuals, a query is at the unit level, where the structural assignments are changed, but the exogenous noise identical that of the observed datum.
(a) Generating Samples for Observational/Iteeeentiii the interventional distribution p(X do(X, := 7)) can be generated as follows. For an intervened node with intervention %. the sampled value is always the intervention value, therefore intervened node i, assume by induction we have the generated parent values 8 Xpa,- To generate X; we first sample the latent vector Z, ~ NN(0, Ia,) where de is the dimension X, Then taking Z, as the noise for node i, we compute X, = Dec, (Z,, X &pa,) as the generated sample value for node i. This value X, is then used as the parent value for the children of node Samples from a DCM model that approximates the observational distribution p(X) can generated by setting
Queries. Samples from DCM model that approximates
generate X, 7/- For non-
0. See Algorithm for the pseudocode.
(b) Counterfactual Queries. Consider a factual observation z" := (zF..... xkk and interventions I with values 7. We use DCM model to construct a counterfactual estimate a CF as follows. The counterfactual estimate only differs from the factual value on intervened nodes or descendants of an intervened node. Similarly to interventional queries, for each intervened node I, .2 == %/- For each con-intervened node that descendant of any intervened node, assume by induction that have the generated counterfactual estimates &CF To obtain C.
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Algorithm 4 Obseevational/miirveettooa Sampling
Input: Intervention set with values ry (I 0 for observational sampling)
:....... K do 2: Zi ~ (0, Ia.)
I: for i .....:
(in topological order)
3: if E I then
4:
X, t %
else
X, t Dec, (Zi; )pa))
end if
end for
9: Return x= (X1....,
Algorithm 5 Counterfactual Estimation
Input:
Intervention set with values y, factual sample xF = (x (xf....,,7) x
for: L,...,K do
(in topological order)
if I then
3:
2CF <-
4:
else if is not descendant of any intervened node in Z then
5:
6: else
of t (nc sapaa)
(abduction step) (action and prediction steps)
8:
F. + Dec,(2)))) (zr pp,,))
9:
end
10:
end for
11:
Return &CF = ( ,..., CF
first define the estimated factual noise aP = Enc, pm, ). Then generate counterfactual estimate by
using ef as the noise for node i, by decoding, .CF == (ef pa, See Algorithm) for the pseudocode.
Note that with FF, we assumed full observability! since because Algorithm produces counterfactual estimate for each node. However, when intervening on X2 and if the only quantity of interest counterfactual on some X*, then you only need factual samples from {X, : X, is on path from Xx - X+} (Saha & Garain (022). In practice, this could be further relaxed by imputing for missing data, which is beyond the scope of this work.
4 Bounding Counterfactual Error
We now establish sufficient conditions under which the counterfactual estimation can be bounded. In fact, the results in this section not only hold for diffusion models, but to more general setting of conditional latent variable models satisfying certain properties. Another feature of this result, is that it also extends, under an additional assump- tion, to the more challenging higher-dimensional case. All proofs from this section are collected in Appendix A
We focus learning single conditional latent variable model for an endogenous node X., given its parents Xpa. the models learned for different endogenous nodes do not affect each other. Since the choice of node plays role, we drop the subscript i in the following and refer to the node of interest as X, its causal parents as Xpa. its corresponding exogenous variables as U, and its structural equation as X := f(Xpa. U). Let the encoding function g : x Xpa + Z and the decoding function h Z x Xps + X, where the latent space. In the DCM context, the functions y and correspond Enc and Dec functions, respectively.
It is well-known that certain counterfactual queries are not identifiable from observational data without making as- sumptions on the functional relationships, even under causal sufficiency (Pearl (0096). Consequently. recent research has been directed towards understanding the conditions under which identifiability results can be obtained (Lu et
This common assumption literature also made the related work egg. Sánchez-Martin
2022 Khemakhem al.
2021
Pawlowski al. 2020).

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2020 Nast- Essssaay & Kiciman| 2023 Nasr- Esfahany al. 2023
Assumption 2 of our Theorem 1 ensures that the true counterfactual outcome is identifiable, see e.g., (Lu et al. 2020 Theorem 1) or (Nasr-Estahany & Kiciman 2023 Theorem 5). In the context of learned structural causal models to determine whether a given counterfactual query can be answered with sufficient accuracy, requires also assumptions on the learned SCM, e.g., encoder and decoder in this
case.
Our first result presents sufficient conditions on the latent variable encoding function and the structural equation under which we can recover latent exogenous variable up to a (possibly nonlinear) invertible function. We start with a one- dimensional exogenous noise U and variable X E X C IR. In Section 4.1 we provide a similar theorem for the higher- dither-dimensional case where E I th for m 2 in Theorem 2 with a stronger assumption on the Jacobian of f and g.
Theorem 1. Assume for X E X' C.
R and exogenous noise U ~ Unif(0, 11, X satisfies the structural equation:
X = s(Xpa; U), where Xpa E Xma C CR" are the parents of node X and U 11 Xpa- Consider an encoder-decoder model with encoding function X' x Xpa + Z and decoding function h 2 x Xps + X, Z 9(X, Xpa); X :
MZ xa).. Assume the following conditions:
1. The encoding is independent of the parent values, xpx, 11 Xpa.
2. The structural equation f is differentiable and strictly increasing with respect to U for all ppa E Xpa-
3. The encoding g is invertible and differentiable with respect to X for all Ppa E Xps.
Then, x(X,Xpx) = =Q(U), for invertible function
Discussion on Assumptions Underlying Theorem
1
(1) Assumption of independence between the encoding and the parent values may appear strong, but is in fact often valid. For example, in the additive noise setting with f(XXpa: U) := f (Xpa)++ +U where Xpm and U are independent, if the fitted model f= f', then the encoder g xpa) = f(Xpa pa: U)- - f(xpa) = U and by definition U is independent
of Xpm 2 The same assumption also appears in other related results on counterfactual identifiability bijective SCMs, see, c.g.. (Nasr-Esfahany
Kiciman 2023). We conduct empirical tests to further confirm this assumption by examining the dependence between the parents and encoding values. Our experimental results show that DCMs consistently fail to reject the null hypothesis of independence. This implics that independent encodings can be found in practice. We provide the details of these experiments Appendix
al.] 2023 Theorem 5.3) and proof of Theorem 5 in (Nasr-Esfahany
F
(2) Assumption 2 is always satisfied under the additive noise model, where f(Xpa. pa: U) f'(X U. It is also satisfied by post non-linear models, under the standard requirements on identifiability as expressed in (Zhang & Hy- varinen 2 Assumption 2 is also satisfied by heteroscedastic noise models (Strobl & Lasko 2023 p Again, the recent results about counterfactual identifiability, e.g., EsfaaaEy & Kiciman 2023 Theorem 5) and (Lu et al. 2020 Theorem 1), also utilize the same assumption.
Through our strictly increasing in U assumption, we obviate distinguishing cases like s(Xpa, U) = Xpm U vs. f(Xpa: U) == Xpa U. which otherwise will be indistinguishable for symmetric distributions U (see also remark below). For example, for any fixed value of Xpa. f(Xpa. U) := Xpm - U is not increasing in U, so such structural other words, among these two cases, this assumption only
equations are automatically eliminated in Theorem permits the additive noise form f f(Xpa U) Xpa U.
"In general, if we have a good approximation ff by some f. then the encoding 9 Xpa) = X (Xpo) would be close to U, as also noted by (Hoyer et al. 20008)
"T encourage independence, one could also modify the original diffiusion model training objective to add a Hilbert-Schmidt independence criterion (HSIC) (Gretion et al. 2007) regularization term. Our experiments did not show clear benefit of using this modified objective, and we leave further investigation here for future work.
*Zhang & Hyvarinen (2012) (see Eqn. 2) defined post-nonlinear models as f(Xpa, U) := (X ++) where Xpa and U are independent. function fi is nonconstant, and fa is invertible. The invertibility assumption on f2 implics that is strictly increasing (or decreasing) in U. Additionally, (Zhang & Hyvarinen 2012) make differentiability of f2 assumption (sec Assumption Al) for identifiability, which implies the f is
differentiable in U.
Heteeoscedastic noise models are generally defined as (Xpa,, U) == f'(Xp..) + ((xo)) U where the function g is assumed to be strictly positive (see Definition in (Strohl Tasko 2023), which makes it compatible with our Assumption of Theorem 1.
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(3) We may consider transformations of the uniform noise U to obtain other settings, for example additive Gaussian noise. For a continuous random variable U' with invertible CDF F and the structural equation F-)), we have U" - (U) and the results similarly hold.
We now discuss some consequences of Theorem for estimating counterfactual outcomes.
now look at condition under which the counterfactual estimate produced by the encoder-decoder model matches the true counterfactual outcome 6 The idea here been, if no in- formation is lost in the encoding and decoding steps, i.e., ((((.. X xpa), xpa) xpa)
1. Perfect Estimation. Using Theorem
X, and assuming Theorem [ (g(X, pa) = q(U)), we have h(q(U)) Xpa) = X = s(Xpa: & (U)). This means that in the abduction step, the encoder-decode model could recover ((U), but in the prediction step, first applies the inverse of n to the recov- ered exogenous variable, and then f. Thus, the counterfactual estimate equals the true counterfactual outcome. We formalize this in Corollary
1 Furthermore, assume the encoder-decoder model pair (g. h) satisfies: h(g( (X, xp xpa) = X. Consider a factual sample pair x" := (p.) where x := f(Ipas u) and an in- tervention do Xps ;= =)) Then, the counterfactual estimate, given by h(g(z, xp)) matches the true counterfactual outcome CF f(7, u))
Corollary 1. Assume the conditions of Theorem
Comparison with Recent Related Work. Recent studies by
(2023) and Nasr-astahany et al. (202)) have explored the problem of estimating counterfactual outcomes with learned SCMs. In particular, Nasr- Esfahany & Kiciman 22023 Theorem 5) consider setting where the SCM X == f(X U) is learned with a bijective et al. 2023 Theorem 5.3) considered a closely related problem of learning a ground-truth bijective SCM. The conditions underlying ours and these results are not directly comparable because, unlike our setup, they do not explicitly consider an encoder-decoder model. Our results provide precise conditions the encoder g and decoder h for recovering the correct counterfactual outcome, and can extend these results to obtain counterfactual estimation error bounds under relaxed assumptions, problem that has not been addressed previously.
& Kiciman
(deep conditional generative) model f(Xpe. pa. Ú)
Counterfactual identifiability in a different context of reinforcement learning was also established by (Lu et al. 2020). Their result relies on incomparable assumptions state-action pairs. Furthermore, our proof techniques quite different from Lu et al. (2020) who rely technique based on analyzing conditional quantile, unlike an algebraic technique employed here.
2. Estimation Error. Another consequence of Theorem | iis that can bound the counterfactual error terms of the reconstruction error of the encoder-decoder model. Informally, the following corollary shows that if the reconstruction h(g(X, Xpp), pa) is "close" to X (measured under some metric d(-,-)), then such provide "good" counterfactual estimates. To the best of our knowledge, this is the first result that establishes bound on the counterfactual error in relation to the reconstruction error of these encoder-decoder models.
models can
Corollary 2. Let Z O. Assume the conditions of Theorem ! Furthermore, assume the encoder-decoder model pair (g, h) under some metric d (e.g.. || (112), has reconstruction error less than т: d(h(g(X, Xpa), Xpa), X) S T. Consider a factual sample pair xF := (x, pa)) where x == fIIpa u) and an intervention do := 7). Then, the error between the true counterfactual CF = f(y, u) and counterfactual estimate given by h(g(z, Zpa)) 7p)). ) is at most T. In other words, d(h(g(x, pppa), 7), CF) I T.
The above result suggests that the reconstruction error can serve as an estimate for the counterfactual error. While the true value of is unknown, we may compute reasonable bound by computing the reconstruction error over the dataset.
4.1 Extension of Theorem to Higher-Dimensional Setting
In this section, we present extension of Theorem I to a higher dimensional setting and use it to provide coun- terfactual identifiability and estimation error results. Since we are now dealing with vector-valued functions, we use Jacobians. Let J fzm denote the Jacobian matrix obtained by evaluating the Jacobian (with respect to U)) of f(X U) at Xpa Tps- Let JgHpa denote the Jacobian matrix obtained by evaluating the Jacobian (with respect to X) of Xpa) at Xpa Ipa-
Note that identifiability of counterfactual outcomes, does not require identifiahility of the SCM.

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Published in Transactions on Machine Learning Research (11/2024)
Theorem 2. Assume for X E X C IR' and continuous exogenous noise U Unif(0, 11Thh for 2 3, and X satisfies the structural equation
x= f(Xpa. U)
(6)
where pm Xpa CRdt are the parents of node X and U 1L Xpa Consider encoder-decoder model with encoding function g X x Nm + Z and decoding function h Z x Xpa + *X
== b(Z, Xpa). Z = g pa),
Assume the following conditions:
1. The encoding is independent of the parent values, g(X, (XXX) Xpa) 11. Xpa-
2. The structural equation f is invertible and differentiable with respect to U, and Sn is p.d. for all upa E Xpa.
3. The encoding g invertible and differentiable with respect X, and Jgspa p.d. for all Tpa Xpa-
4. The encoding 4= (U) 9 (r pp)) satisfies J q lapi. (s) c(Epa) for all 2 and Tpa E Kpa- where e. is scalar function and A is an orthogonal matrix.
Then, g(f Xpa) [() for invertible function
The interpretations behind Assumptions 1, 2, and in Theorem 2 are similar to corresponding assumptions in The- orem Assumption 4 however technical, and will explain the need for it below. In Corollaries 3 and 4 (Appendix A. restate Corollaries D and 2 to this higher-dimensional setting. The proofs are identical to that of Corollaries and 2
with the only change being that the role of Theorem I in those proofs is now replaced by Theo-
rem
On Negative Result of Nasr-Esfabany Kiciman (0023).
Nasr-Esfahany Kiciman 20233 presented gen- eral counterfactual impossibility identification result under multidimensional exogenous noise. The construction in Nast-Esfahany & Kiciman (2023) considers two structural equations f. that are indistinguishable in distribution. Formally, let R E IR be rotation matrix, and E RT be a standard (isotropic) Gaussian random vector. Define,
[(Xpar par U) f(Xpar R- R-
for Xpa A for Xpa B
('(Xpa U)
(8)
where the domain Xpa is split into disjoint and B. Now, f and f' generate different counterfactual outcomes, for counterfactual queries with evidence in A and intervention in B (or the other way around).
In Theorem - we avoid this impossibility result by assuming that we can construct an encoding of a "special" kind captured through our Assumption 4. particular, consider the encoding azy. at a specific parent value Tpa as a function of the exogenous noise U. The assumption states that the Jacobian of the encoding is equal to c(*pa) A for scalar function and orthogonal matrix A. However, is important acknowledge that this assumption is highly restrictive and difficult to verify, not mention challenging to construct in practice with just observational data. Our intention is that these initial ideas can serve as a starting point for addressing the impossibility result, with the expectation that subsequent results will further refine and expand upon these ideas.
5 Experimental Evaluation
In this section, we evaluate the empirical performance of DCM for answering causal queries on both synthetic and real world data. Our primary objective through these experiments is not to tackle specific causal inference problem, but to demonstrate that our DCM approach provides unified and flexible framework for answering wide range of obser- vational, interventional, and counterfactual queries. For the real data experiments, we consider an interventional data experiment on fMRI data (Section 5.2) and an experimental study for the problem of estimating individual treatment effects (Appendix
"oode for reproducibility: http:///www.../ooooooooooooo/taa
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Diffusion Model Implementation and Training. For our implementation of the Ee model in DCM, we use sim- ple fully connected neural network with three hidden layers of size [128, 256, 256] and SiLU activation (Elfwing et al. 2018). We fit the model using Adam with a learning rate of le-4, batch size of 64, and train for 500 epochs. Since the root nodes lack parents, the only form of counterfactual reasoning involves directly intervening on the root node, which may be done trivially. Therefore, for root nodes, we do not train diffusion models and instead sample from the empirical distribution of the training data. Additional details about the diffusion model parameters are in Appendix C.I
Compared Approaches. For fair comparison, our evaluation centers on methodologies that allow for both inter- ventional and counterfactual estimation. With this criteria, we primarily compare DCM to two recently proposed state-of-the-ars schemes VACA (Sánchez-Martin al. 2022) and CAREFL (Khemakhem et al. 2021), and a general regression model that assumes an additive noise model which we refer to as ANM For VACA and CAREFL, we use the code provided by their respective authors. The ANM approach performs model selection over a variety of models, including linear and gradient boosted regressor, and we use the implementation from the popular DoWhy causal infer- ence package (Sharma al. 2019 Blöbaum et al.) 2022). Additional details how ANM answers causal queries are provided in Appendix C.2 and implementation details for VACA, CAREFL, and ANM are in Appendix C.1
5.1 Synthetic Data Experiments
For generating quantitative results, we use synthetic experiments since we know the exact structural equations, and hence we have access to the ground-truth observational, interventional, and counterfactual distributions. We present results on two larger graphs here, and defer results on a of four smaller graphs Appendix [D2 In Appendix [D3 we also present results by generating synthetic data based on the real world graph from the Sachs dataset (Sachs et al. 2005).
The two graphs, considered here, include a ladder graph structure (see Figure D and a randomly generated graph structure. Both graphs are comprised of 10 nodes of three dimensions each, and the random graph is a randomly sampled directed acyclic graph (see Appendix C for more details). Since each diffusion model only uses the parents as input, modeling each node depends only on the incoming degree of the node (the number of causal parents).
Following Sänchez-Martin et al. (2022), for the observational and interventional distributions, we report the Maxi- mum Mean Discrepancy (MMD) Gretton et al., 2012) between the true and estimated distributions. For counter- factual estimation, we report the mean squared error (MSH) of the true and estimated counterfactual values. Again following Sarchez-Martiin al. 22022), we consider two broad classes of structural equations:
Additive Noise Model (NLIN): fi(Xpa, Ut) f'(ppa)) + Ug. In particular, we will be interested in the case where fi's are non-linear.
2. Nonadditive Noise Model (NADD): fi(X U.) is an arbitrary function of Xpa, and U,.
To prevent overfitting of hyperparameters, we randomly generate these structural equations for each initialization. Each structural equation is comprised neural network with single hidden layer of 16 units and SiLU activation Ellwwng et al. 2018) with random weights from sampled from [-1.
Each simulation generates n 5000 samples as training data. Let M be a fitted causal model and M be the true causal model, both capable of generating observational and interventional samples, and answering counterfactual queries. Each pair of graphs and structural equation type is evaluated for 20 different initialization, and we report the mean value. We provide additional details about our observational, interventional, and counterfactual evaluation frameworks in Appendix C.4
Synthetic Experiments Results. In Table D we provide the performance of all evaluated models for observational, interventional, and counterfactual queries, averaged over 20 separate initializations of models and training data, with the lowest value in each row bolded. The values are multiplied by 100 for clarity. We also provide boxplots of the performances in Figure (Appendix
We see DCM and ANM are the most competitive approaches, with similar performance on observational and inter- ventional queries. If the ANM is the correctly specified model, then the ANM encoding should be close to the true
IIn spite of best efforts, we were unable run a proper comparison against the very recent approach proposed by Javaloy et 2023 due challenges in adapting their code into our settings.
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DCM
ANM
VACA
CAREFL
SCM
Metric
(x10-3)
(x10
(x10
(x10-3)
Obs. MMD
0.44:10.14
0.63+:021
2.82::0.33
13.41=1.14
NLIN
Int. MMD
1,63::0,00
1.80:0.17
4,48:10.78
15.01=11.23
Ladder
CF MSE
3,42:11.67
10.65-+2.48
41.03-+:1900
17.46:16.04
Obs. MMD
0.32
0,4010.16
3.22:11055
14.60=1.33
NADD
Int. MMD
1.54
1.57:0.15
5.13#1.16
1..87
CF. MSE
4.28:+2.99
10.71
27.42:112.34
22.26=11.75
Obs. MMD
0.28
0.47
1.82+0.73
12.11=1.21
NLIN
Int. MMD
1.45
1.88+0222
3.52
14.15=2.34
Random
CF. MSE
9.511113.12
23.68-28.49
82.10:178.49
52.57:182.03
Obs. MMD
0.19:10.05
0.31
2.09110.00
12.63
NADD
Int. MMD
1.42:+0225
1.73+0.44
1.24
14.65=1.77
CF. MSE
50.13
44.76=26.02
124.82=275.09
54.29=13.66
Table 1: stadaard deviation of observational, interventional, and counterfactual queries of the ladder and ran- dom SCMs in nonlinear and nonadditive settings over 20 random initializations of the model and training data. The values are multiplied by 100 for clarity.
encoding, assuming the regression model fit the data well. We see this in the nonlinear setting, ANM performs well but struggles to outperform DCM, perhaps due to the complexity of fitting a neural network using classical models. Note thats since observational and interventional queries are inherently easier than counterfactual queries, is natural that we observe smaller improvements over other baselines.
Our proposed DCM method exhibits superior performance compared to VACA and CAREFL, often by much order of magnitude. The better performance of DCM over CAREFL may be attributed to the fact that DCM uses the causal graph, while CAREFL only relies on a non-unique causal ordering. In the case of VACA, the limited expressiveness of the GNN encoder-decoder might be the reason behind its inferior performance, especially when dealing with multivariable, multidimensional complex structural equations as considered here, a shortcoming that has also been acknowledged by the authors of VACA (Sánchez-Martin et al., (022). Furthermore, VACA performs approximate inference, e.g. when performing counterfactual query with do X1 = 2), the predicted counterfactual value for X, is not exactly 2 due to imperfections in the reconstruction. This design choice may result in downstream compounding errors, possibly explaining discrepancies performance. To avoid penalizing this feature, all metrics are computed using downstream nodes from intervened nodes. To evaluate computational efficiency, DCM with VACA and CAREFL using single experimental setup (see Appendix D.Ipp The results suggest that DCM is significantly more computationally efficient than both VACA and CAREFL, with training taking 7 times less time. Lastly, the standard deviation of DCM is small relative to the other models, demonstrating relative consistency, which points to the robustness of our proposed approach.
compared
5.2 Real Data Experiments
We evaluate DCM interventional real world data by evaluating our model the electrical stimulation interven- tional fMRI data from (Thompson et al.) 2020), using the experimental setup from (Khemakhem et al.) (021). The IMRI data comprises samples from 14 patients with medically refractory epilepsy, with time series of the Cingulate Gyrus (CG) and Heschl's Gyrus (HG). The assumed underlying causal structure is the bivarate graph CG + HG. Our interventional ground truth data comprises an intervened value of CG and an observed sample of HG. We defer the reader to (Thompson et al. 2020 Khemakhem et al. 2021) for more thorough discussion of the dataset.
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Algorithm
Median Abs. Error
Mean Abs. Error
DCM
0.5981 + 9.5e-3
0.5779 + 1.9e-3
CAREFL
0.5983 + 3.0e-2
0.6004 + 2.2e-2
ANM
0.6746 + 4.8e-8
0.6498 + 1 1.4e-6
Linear SCM
0.6045 + 0
0.6042 + 0
Table 2: Performances for interventional predictions on the fMRT dataset, of the form median/mean absolute standard deviation, using the mean over 10 random seeds. We do not include VACA due to implementation difficulties. The results for CAREFL, ANM, and Linear SCM are consistent with those observed by (Khemakhem al. 2021). We note that the Linear SCM has zero standard deviation, the ridge regression model does not vary with the random seed.
In Table we note that the difference in performance is more minor than our synthetic results. We believe this due to reasons. Firstly, the data seems inherently close linear, exhibited by the relatively similar performance with the standard ridge regression model (Linear SCM). Secondly, we only have a single ground truth interventional value instead of multiple samples from the interventional distribution. As result, we can only compute the absolute error based on this single value, rather than evaluating the maximum mean discrepancy (MMD) between the true and predicted interventional distributions. Specifically, the above table, we compute the absolute error between the model prediction and the interventional sample for each of the 14 patients and report the mean/median. The availability of a single interventional introduces possibly large amount of irreducible error, artificially inflating the error values. For more details on the error inflation, see Appendix C.3
6 Concluding Remarks
We demonstrate that diffusion models, particular, the DDIM formulation (which allows for unique encoding and decoding) provide flexible and practical framework for approximating interventions (do-operator) and counterfactual (abduction-actionn patioonaaattioo--eitttioon steps. Our approach, DCM, is applicable independent of the DAG structure. We find that empirically DCM outperforms competing methods in all three causal settings, observational, interventional, and counterfactual queries, across various classes of structural equations and graphs.
While not in scope of this paper, the proposed DCM approach can also be naturally extended any setting where diffusion models applicable like categorical data, images, etc. For higher dimensional spaces, believe DCMs should scale nicely, as diffusion models are typically deployed for nigh-dimensional image settings and exhibit SOTA performance. Furthermore, we may leverage many of the optimization and implementation tricks, as diffusion models are a very active field research.
The proposed method does come with certain limitations. For example, with all the previously mentioned related approaches, DCM precludes unobserved confounding. The theoretical analyses require assumptions, not all of which are easy to test. However, our practical results suggest that DCM provides competitive empirical performance, even when some assumptions needed for our theoretical guarantees are violated.
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arXiv preprint
A Missing Details from Section 4
Notation. For two sets X, a map f 2 + ), and a set S C X, we define f(S) = (f(x) E For x E X, we define x + = {ar + +' x' E XJ. For a random variable X, define px (x(x) as the probability density function (PDF) at z. We use p.d. to denote positive definite matrices and fll to denote the Jacobian of f evaluated at x. For function with two inputs f(-,),, ), define fa(Y) == f(r and (x) = f(X.
Lemma 1. For U1 E, consider a family of invertible functions qum - for Ipa E Xpa C. Rd then
dgrps (z)) c(2) for all xpa Npa if and only if 9zma can be expressed as du
azza (u) = 4(u + r((pa))
for some function r and invertible q.
Proof. First for the reverse direction, we may assume Izpa (u) = q(u + r(ppa)). Then
dgzpa (u)
dg
(u + r rxxpa))
du
du
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Now plugging in qapa (2) 9 r(ppa))
dgzna du
dq (q (z) - r(ppa) + r(ppa))
dg du
(9 (z))
(q (z)) = c(z).
du
Therefore
dgapa
du (gap (z)) is a function of z.
dgmpa (9 (2)) c(z). Define 8mp : - 24 to be the inverse of am By the du
For the forward direction, assume inverse function theorem and by assumption.
d8
dg dg (2)
(2)
dz
dz
dgupa du
(g (2))
c(2)
for all paa- Since the derivatives of 8xpa are equal for all pas by the mean value theorem, all Sup= are additive shifts of each other. Without loss of generality, we may consider an arbitrary fixed Vapg Xna and reparametrize 8xpa
Surpa (2) Sirpao (2) r (Zpp))
Let Szpa (2). Then we have
pxp (2) + (xppa)
qup (u) 2 q2 (u + r(Fpa)):
and we have the desired representation by choosing q Qupao"
Theorem 1. Assume for X E X C Ph and exogenous noise U Unif(O, 1), X satisfies the structural equation:
X = f(Xpar U), where Xpa E Kpa C Rdd are the parents of node X and U 1L Xpa. Consider encoder-decoder
model with encoding function g X Xps= - 2 and decoding function h 2 x Xps + X, 7 = 9(X, X.XXXpall Xpa); X
h(Z, ppa). Assume the following conditions:
1. The encoding is independent of the parent values, g(X, Xpa) 11. Xpa-
2. The structural equation f is differentiable and strictly increasing with respect to U for all Tpa E Xpa.
3. The encoding g invertible and differentiable with respect X for all zaaa E Xpa.
Then, g(X, Xpa) = @(U) ) for an invertible function q.
Proof. First, we show that g(X, gXf( xpa is solely a function of U.
Since continuity and invertibility imply strict monotonicity, without loss of generality, assume g is an strictly increasing function (if not, we may replace with -g and use h(-2, Xpa)). By properties of the composition of functions, ar (U) = pTpp Epa) is also differentiable and strictly increasing with respect to U. Also, because of strict monotonicity also invertible.
By the assumption that the encoding Z is independent of X pa.
2 -
Yops (U) 11. X pa-
Therefore the conditional distribution of Z does not depend on Xpa. Using the assumption that U JL Xpa. for all Ta Npa and in the support of Z, by the change of density formula,
Pu (q (2))
1 19 (2) E [0, = (z).
Pz(z) =
pa
(10)
dgipp
du (g (2))
dgsna du (nonh (z))
The numerator follows from the fact that the noise is uniformly distributed. The term duspa (g= is nonnega- du
tive since 2r... is increasing. Furthermore, since pz(z) 0, the numerator in Eq. 10 is always equal and the
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denominator must not depend on Xpa
dg du (9 (2)) = c2(2)
for some function c2. From Lemma (by replacing a by ppa). we may express
axga (u) = q(u + r(ppa))
(11)
for an invertible function q.
Next, since Z TT xa. the support of Z does not depend Xa. equivalently the ranges of qu, and gx, equal for all æ1, 22 E pp.
Qx. (10, 1J) = 9x, (10,
(12)
Applying Eq. 11 and the invertibility of
<(0,, ( [[00 r((z))
[0,1]+r(z,) [0,1] r(r2)
(r())),((1))+11 (r(22),r(x)) +1]
Since this holds for all æ1, 22 E Xpa. have r( is constant function, or r(Zpa) Thus let be g(u)
q(u + r) qupa (u), which is solely a function of U for all Tpa. For all Tpa.
g(f U), xpa)) q2pa (U) = q(U) = g(f(Xpa. Upa) = =Q(U),
(13)
for an invertible function q. This completes the proof.
Corollary 1. Assume the conditions of Theorem 1 Furthermore, assume the encoder-decoder model pair (g. h) satisfies: h(g( (X, Xpa); Xpa) X. Consider factual sample pair xF (z, xpa) where x := f( u) and an in-
tervention do X pa == 7). Then, the counterfactual estimate, given by h(g(2, (p) matches the true counterfactual outcome x CF f u).
Proof. For the intervention do(Xpa pa -y). the true counterfactual outcome is xF := fly, u). By assumption, h(g(xCF Ippa), paa = CF, Now since Eq. 13 holds true for all Xpa and U, also holds for the factual and counter- factual samples. We have,
g( xpa) ((f u), q(u) g(f(r, u), = OF ,)).
Therefore, the counterfactual estimate produced by the encoder-decoder model
h(g(t, ())),,) ((g(x .7),7) =x CF
This completes the proof.
Let 2 O. Assume the conditions of Theorem ! Furthermore, assume the encoder-decoder model pair (g, h) under some metric d (e.g.. || [|2), has reconstruction error less than т: (o,(g(XXXpp)))) Xpa),Xppa Xpa), X) S т. Consider a factual sample pair F == (x, Ppa) where := fI u) and an intervention do(Xps. := 7)- Then, the error between the true counterfactual CF = f(y, u) and counterfactual estimate given by h(g(z, xpa): is at most T. In other words, d(h(g((,, F.pa), 7), CF) <
Corollary 2.
Proof. For the intervention do(Xpm : 7), the counterfactual outcome is x := f(y, u). Since Eq. 13 holds true for all Xpa and U, it also holds for the factual and counterfactual samples. We have,
g(z, xpa) (I u), Ipa)
q(u) g(f(r,)) CF
(14)
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By the assumption the reconstruction error of the encoder-decoder and 14
d(hlglæ CF 7), 7), T
(15)
I(p( 7), CF
S.
(16)
We discuss lemma that is an extension of Lemma to higher dimensions.
Lemma 2. For U, 2 C Rmm, consider family of invertible functions qs u - 2 for Ipa E Xpa C IR., if qap (9rpa 1 (z)) = cAAfor for all Ipa E Xpa then qu can be expressed as
ap (u) = q(u r(Zpa))
for some function v and invertible 4-
Proof. Assume Japp (T (=)) = c.A. By the inverse function theorem,
J J 12
J lo (x)
(17)
Define Smpa : to be the inverse Gapa By assumption
det J qupa Ma det J det J 2 eppal cA
for all pa and a constant e and orthgonal matrix A. Since the Jacobian of 8xpm a scaled orthogonal matrix, 8z, is a conformal function. Therefore by Liouville's s theorem, Szpa is a Möbius function (Blair 2000), which implies that
gu (z) bapa + Chir Auza - Ozpa)/ 3/1lz agpaa ||
(18)
where Arya is orthogonal matrix, & E 2), axpo R and Csrp E The Jacobian of 8xya is equal to cA by assumption
Okupp Appa argaall 1|5
T (1- (2 axpa (2 axpa )xpu))
J8ppal
cA.
|| axpa 112
This imposes constraints on variables a, a, and s. Choose such that = axpa = kv for a unit vector v and multiply by A-1
CAA-1 Ap =
Ossp
I --
|kou||
k-2 11:11112
ck
= EVUT +
AA-1 ppa
axr
I 2, choosing different values of kt implying different values of 2, results in varying values of the right hand side, which should be the constant identity matrix. Therefore we must have = 0. This also implies that agpa = and = Aspa This gives the further parametrization
Szpa (2)
bron - -CA(z- Ozpa) b' p'(xpa) + cAz
where b' b'(xpa) = bapa - cAa
Without of loss of generality, we may consider arbitrary fixed Zpap E Xpas
8xyo0 (2) -erpa (2) r(pppa) 1' (2(pp))) - D' (xpa))-
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Let Sxpa (2). Then we have
pxp00 (z) = + (((ppa)
aap (1) = pao (u + r r(Fpa));
and we have the desired representation by choosing o Impoo"
Theorem 2. Assume for E & C I and continuous exogenous noise U Unif 11 for m 2 3, and X satisfies the structural equation
X (XXpa U)
(6)
where pa E Xpa C Rdd are the parents of node X and U 1L. Xpa- Consider an encoder-decoder model with encoding
function g : X x Xpa- + and decoding function Z Xpa- + XX,
x= 1(2, Xma) :: (==g(X,Xpi))
(7)
Assume the following conditions:
1. The encoding is independent of the parent values, g(X, xpa) 11. X pa-
2. The structural equation f is invertible and differentiable with respect to U, and Jn m is p.d. for all Tp E Xpa-
3. The encoding is invertible and differentiable with respect to X, and Jg is p.d. for all Tpp E Xpa-
4. The encoding qupa (U) = (I U), pa) satisfies Jqpp (m (6) = p( for all E Z and Tps E Aps. where is scalar function and A is orthogonal matrix.
Then, (Xpa; = ((U))oo for an invertible function q.
Proof. We may show that XX,) Xpa) = 9( (o[[(Xa;;0) U), X xpa) is solely a function of U.
By properties of composition of functions, qapa (U) = g (xpa U), Tppa) is also invertible, differentiable, Since Ja and Jgrpa are p.d. and Jqzp- f J9 then Jqpp p.d. for all zpa E Xpa as well.
By the assumption that the encoding 7 is independent of pa
Z 9Xpm (U) 11 X pa.
(19)
Therefore the conditional distribution of Z does not depend on Xpa. Using the assumption that U II X Xpa. for all pa Kpa and in the support of Z, by the change of density formula,
(z(2)= = PU (9 (2))
2 m (2)
(z).
(20)
det J Jgr 4:
det J Jqzp Iagia (z)
The numerator follows from the fact that the noise is uniformly distributed. The determinant of the Jacobian term Jqsp (9mp (2)) nonnegative since Jysp. is p.d. Furthermore, since pz(z) > O, the numerator in Eq. 20 is always equal to 2-m and the denominator must not depend on Xpa-
det Jazp (ight = (2((2)
(21)
for some function c2. From our assumption, Jqpp wipl = c(Ipa) ) for an orthogonal matrix for all 2. Applying
this Eq. 21
det Jq qopa (2) det (eetc((paa)a c(Epa) o2(2).
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which implies c(Epa) is constant function, J qspa quiaa (2)
c.. By Lemma may express qspa (u) as
an (u) - q(u T(Upa))))
(22)
for an invertible function q-
Next, since Z 11. X pa. the support of Z does not depend on X Xpa equivalently the ranges of Oeper and Orpaz are equal for all 1 22 E Xpa.
az (10 = 92, (10,
(23)
Applying Eq. 22 and the invertibility of q,
(100, 11 r(x1)) = ((00, r((2)) 10, r(()) [0. 1|"H + r(12)
(r(2)),(Z1)) + 1| [r(m2), + 1Jmh.
Since this holds for all æ1, 22 E Pas we have r(z) is a constant, or r(x) Thus be = ((u+r) = Axpa (u),
which solely a function of for all paa For all Taa.
g(f(Fpa U, xpa)) = Azya (U) = ((U) = g(f U), Xpx Q(U).
(24)
This completes the proof.
Corollary 3. Assume the conditions of Theorem satisfies
Furthermore, assume the encoder-decoder model pair (g. h)
hgg (X, pm), Xpa) = X.
(25)
Consider a factual sample pair (x, xppa) where == pa u) and an intervention do (o(xpa == 7). Then, the given by h(g(z, ; 7) matches the true counterfactual outcome x = -(7, u).
Corollary 4. Let y 2 0. Assume the conditions of T'hhorrm Furthermore, assume the encoder-decoder model pair (g. h) under some metric d (e.g., I12),, has reconstruction error less than T,
d(h(g(X, Xpa), XI) ST.
(26)
Consider a factual sample pair (x,, Tppa where x = [Tipa u) and an intervention do(Xpa = 7). Then, the er- ror between the true counterfactual CF = f(y, u) and counterfactual estimate ((glI,Tp)) 7) at most T, i.e. d(b(g(x, Tpaa) xF S T.
B Testing Independence between Parents and Encodings
We empirically evaluate the dependence between the encoding and parent values. We consider a bivariate nonlinear SCM X; - X₂ where X₂ f(X,, U2) X? + U2 and X1 and U2 are independently sampled from standard normal distribution. We evaluate the HSIC between X, and the encoding of X2- We fit model samples and evaluate the HSIC score 1000 test samples from the same distribution. We compute a p-value using kernel based independence test and compare our performance to ANM, a correctly specified model this setting (Greton et al. 2007). We perform this experiment 100 times. Given true independence, we expect the p-values to follow a uniform distribution. In the table below, we show some summary statisties of the p-values from the 100 trials, with the last row representing the expected values with true uniform p-values (which happens under the null hypothesis).
5000
We provide p-values from the correctly specified ANM approach which are close to a uniform distribution, demon- strating that is possible have encodings that are close to independent. Although the p- p-values produced by our DCM approach not completely uniform, the encodings do not consistently reject the null hypothesis of independence.
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Mean
Std. Dev
10% Quantile
90% Quantile
Min
Max
DCM
0.196
0.207
0.004
0.515
6e-6
0.947
ANM
0.419
0.255
0.092
0.774
3e-5
0.894
True Uniform (null)
0.500
0.288
0.100
0.900
le-2
0.990
Table 3: Table of p-value descriptions.
These results demonstrate that it is empirically possible to obtain encodings independent of the parent variables. We further that the ANM is correctly specified in this setting and DCM is relatively competitive despite being far more general.
C Missing Experimental Details
In this section, we provide missing details from Section
C.1 Model Hyperparameters
For all experiments in evaluation, we hold the model hyperparameters constant. For DCM, we use T 100 total time steps with linear 3r schedule interpolating between le-4 and 0.1, or B, = (0.1 - 10 T-T + 10-4 for i. E [7]. To incorporate the parents' values and time step I, we simply concatenate the parent values and 1/7 as input to the Eg model. We found that using the popular cosine schedule (Nichol & Dhariwal 2021) resulted in worse empirical performance, as well as using a positional encoding for the time f. believe the drop performance from the positional encoding is due the low dimensionality of the problem since the dimension of the positional encoder would dominate the dimension of the other inputs.
1-ع
We also evaluated using classifier-free free guidance (CFG) (Ho & Salimans 2021 to improve the reliance on the parent values, however, we found this also decreased performance. We provide a plausible explanation that can be explained through Theorem I With Theorem [ we would like our encoding g(Y, X) to be independent of X, however using CFG encoding (1 (Y,(Y, X) wY, would only serve to increase the dependence of g(Y, XX) on X, which is counterproductive to our objective.
For VACA, we use the default implementatior training for 500 epochs, with a learning rate of 0.005, and the encoder and decoder have hidden dimensions of size 8, 8] and [16] respectively, a latent vector dimension 4, and parent dropout rate of 0.2.
For CAREFL, we also use the default implementation" T0 with the neural spline autoregressive flows Durkan et al. 2019), training for 500 epochs with a learning rate of 0.005, four flows, and ten hidden units.
For ANM, we also use the default implementation to select a regression model. Given set fitted regression models, the ANM chooses the model with the lowest root mean squared averaged over splits of the data. The ANM considers the following regressor models: linear, ridge, LASSO, elastic net, random forest, histogram gradient boosting, support vector, extra trees, k-NN, and AdaBoost.
C.2 Details about the Additive Noise Model (ANM)
For a given node X; with parents Xpa. consider fitting a regression model f, where fi(Xpa, X₁. Using this regression model and the training dataset is sufficient for generating samples from the observational and interventional distribution, well as computing counterfactuals.
Samples. Samples constructed in topological order. For intervened nodes, the sampled value is always the intervened value. Non-intervened root nodes in the SCM are sampled from the empirical distribution of the training set. new sample for X; is generated by sampling the parent value Xpa, inductively and sampling Ú from the empirical residual distribution, and outputting X; = fx(p),.) + Ū.
Observational/ntereentooo
htttps://githii http:///gtttiii.ccm////mmmmrrrrr
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Counterfactual Estimation. For a factual observation xº and interventions nodes I with values y, the counterfac- tual estimate only differs from the factual estimate for all nodes that are intervened or downstream from an intervened node. We proceed in topological order. For each intervened node i, _. = 7/-- For each non-intervened node downstream from an intervened node, define ui == = - fi( (= pa,)) the residual and estimated noise for the factual sample. Let &CP be counterfactual estimates of the parents of X,. Then ,."h == f(())T)) + a.-
pa,
Therefore, for counterfactual queries, if the true functional equation fa an additive noise model, then if f fi. the regression model will have low counterfactual enoor. In fact, if f. ft. then the regression model will have perfect counterfactual performance.
C.3 Details about Random Graph Generation
The random graph is comprised of ten nodes. We randomly sample this graph by generating a random upper tri- angular adjacency matrix where each entry in the upper triangular half is each to 1 with probability 30%. We then check that this graph is comprised of single connected component (if not, we resample the graph). For a graphical representation, provide example in Figure
C.4 Query Evaluation Frameworks for Synthetic Data Experiments
Observational Evaluation. We generate 1000 samples from both the fitted and true observational distribution and report the MMD between the two. Since DCM and the ANM use the empirical distribution for root nodes, we only take the MMD between nonroot nodes.
Interventional Evaluation. We consider interventions of individual nodes. For an intervention node i, we choose 20 intervention values 7i.-..> 720- linearly interpolating between the 10% and 90% quantiles of the marginal distribution of node i to represent realistic interventions. Then for each intervention do(X; := 7g), generate 100 values from the fitted model and true causal model, X and X for the samples from the fitted model and true model respectively, Since the intervention only affects the descendants of node i, we subset X and X to include only the descendants of node i, and compute the MMD on X and X to obtain distance ōg,j between the interventional distribution for the specific node and interventional value. Lastly, we report the mean MMD over all 20 intervention values and all intervened nodes. For the ladder graph, we choose intervene on X₂ and X3 as these are the farthest nodes upstream and capture the maximum diffieulty of the intervention. For the random graph, randomly select three non-sink nodes to intervene on. A formal description of our interventional evaluation framework is given in Algorithm 6
Algorithm 6 Evaluation of Interventional Queries
1: for each intervention node i do
2: "....., 120 linearly interpolate 10% to 90% quantiles of node
3: for each intervention "/j generated above do
4: Intervene do(X; := 7)))
5: Generate 100 samples X from M and X from M of descendants of
6: ōi, + M))(X.x*)
7: end for
8: end for
9: Output mean of all 8t,j
Counterfactual Evaluation. Similarly to interventional evaluation, consider interventions of individual nodes and for node i, we choose 20 intervention values 7,..., 720, linearly interpolating between the 10% and 90% quantiles of the marginal distribution of node i to represent realistic interventions. Then for each intervention do(X; == 7;). we generate 100 nonintervened factual samples xF and query for the estimated and true counterfactual values cCF and CF respectively. Similarly before, ,CF and CF only differ the descendants of node i, therefore we only consider the subset of the descendants of node i. We compute the MSE ōi,j. since the counterfactual estimate and ground truth are point values, giving us an eтror for a specific node and interventional value. Lastly, we report the mean MSE over all 20 intervention values and all intervened nodes. We use the same intervention nodes as in the interventional evaluation mentioned above. formal description of our counterfactual evaluation framework is given in Algorithm 7 (Appendix C).
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Algorithm 7 Evaluation of Counterfactual Queries
1: for each intervention node i do
2: "l...., Y20 linearly interpolate 10% to 90% quantiles of node
3: for each intervention "ij generated above do
4: Generate 100 factual samples ,i....,, x
Intervene do(X; := 7ty)
Using (x: k== (-=)) compute counterfactual estimates ( =k=1 and true counterfactuals (
k== for all
descendants of
7: ōi,j + M JR=1: (
(R=1)
8: end for
9: end for
10: Output mean of all وú.j
C.5 Explanation of Error Inflation in fMRI Experiments
In our fMRI experiments, we compute the absolute error on a single interventional sample and compute the absolute value. As an intuition for why we cannot hope to have errors close to zero and for why the errors are relatively much closer together, consider the following toy problem.
Assume X1. Xn, MI.....,Y. Yn N(0, 1) and we observe X1. , Xn data. Consider the two following statisti- cal problems:
Learn distribution 1) such that samples from D and samples Y1,..., Y, achieves low MMD.
Learn an estimator that estimates Y1 well under squared error.
In the first statistical problem, if î a reasonable estimator, we should expect that more data leads to lower a MMD, for example the error may decay at a 1/n rate. We should expect MMD values close to zero, and the magnitude of the performance is directly interpretable.
In the second statistical problem, under squared error, the problem is equivalent to
min (Y c)² = min 0 + - 2cf + c
The optimal estimator is the mean 0 and achieves a squared error of 1. Of course in practice we do not know 0, if we were to use the sample mean of X1,.... , X,, we would have an error of the order + 1/n. While we may still compare various estimators, e.g. sample mean, sample median, deep neural network, all the losses will be inflated by 1, causing the difference in performances to seem much more minute.
Our synthetic experiments hope directly estimate the interventional distribution and computes the MMD between samples from the true and model's distributions, implying they are of the first statistical problem. The fMRI real data experiments aim to estimate single intervention value from distribution, implying they are of the second statistical problem. We should not expect these results to be very small, and the metric values should all be shifted by an intrinsic irreducible error.
D Additional Experiments
D.1 Running Time Experiments
We present the training times in minutes for seed the ladder graph using the default implementation and parameters. For a fair comparison, these are all evaluated on a CPU. Note that ANM is the fastest as it uses standard regression models, and our proposed DCM approach is about 7-9x faster than CAREFL and VACA. The generation (inference) times for all the methods are in the order of 1 second. For VACA and CAREFL, we use the implementation provided by the respective authors.
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Nonlinear Equations Observational MMD
Nonadditive Equations Observational MMD
0.14
0.175
0.150
0.12
0.125
0.125
MMD
0.100
MMD
0.125
0.100
0.075
0.075
0.050
0.04
0.050
0.030
0.02
0.025
0.000
0.00
0.000
0.000
Interventional MMD
Interventional MMD
0.20
0.175
0.175
0.150
0.15
0.125
MMD
0.125
MMD
0.15
0.100
0.100
0.10
0.075
0.10
0.075
0.050
0.05
0.05
0.050
0.025
0.025
0.000
0.00
0.00
0.000
Counterfactual MSE
Counterfactual MSE
12.5
0.8
0.5
0.6
0.4
MSE
0.3
0.2
5.0
2.5
0.0
0.0
DCM
ANM VÁCA CAREFL
DCM
ANM VACA CAREFL
DCM ANM CAREFL
DCM ANM VACACAREFL
Ladder
Random
Ladder
Random
Figure 1: Left: Nonlinear setting (NLIN), Right: Nonadditive setting (NADD), Box plots of observational, interven- tional, and counterfactual queries of the ladder and random SCMs over 20 random initializations of the model and training data.
DCM
ANM
VACA
CAREFL
Training Time (in minutes)
15.3
4.3
142.8
110.5
Table 4: Table of training times for single run.
D.2 Additional Synthetic Experiments
In this section, we present additional experimental evidence showcasing the superior performance of DCM in address- ing various types of causal queries. We consider four additional smaller graph structures, which call the chain, triangle, diamond, and Y graphs (see Figure 9.
The exact equations are presented in Table These functional equations were chosen to balance the signal-to-nois ratio of the covariates and noise to represent realistic settings. Furthermore, these structural equations were chosen after hyperparameter selection, meaning did not tune DCM's parameters tune the structural equations after observing the performance of the models.
Consider a node with value X with parents Xpa and exogenous noise U where Xpa. 1L U, and corresponding func- tional equation f such that
X S(Xpa. U)-
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SCM
Nonlinear Case
Nonadditive Case
Chain
X1)
exp(X1/2) + U//4
+X(( 0.5)
fs(Ug, X2)
(X2 - 515 +Us
VX2 X2 + +0/l1/(0.11111 + X2)
Triangle
X2)
2X7 + UU2
XI/((U2 + X1)² +1) U2/4
X1,
20/(1 + exp( + X1)) + Us
(|Ua) + + (x + X2/2+ 13/55)2
X2)
x7 U2/2
(IXi( 0.1)/2 |X,] U2/5
Diamond
f11
X3 - 2/(1 exp(-x1)) + U3/2
1/(1+ + ([Us) + 0.5) exp( -xx + X1))
12U,,,2,,4 X2)
Xa/(IX2 +
+ X3 0.5) U4/10
(X3 + 12-X2+ U4/4 7)2 20
- X2)) X3- + U3/2
(X1 2X2 - 2)( 0.2)
Xz(
20/(1 + cxp((33/2 Xz)) +
Table 5: The equations defining the data generating process in the NLIN and NADD cases.
Further assuming the additive noise model, X == fi(Xpa) + f2(U). - In this additive setting, since Xpa. 11 U, we have
Var [X] = Var [Xi + Var [U)()
We choose f1 and fz such that
Var [f2(U)]
0.05 <
Var [fr [Xpal 0.5,
representing the fact that the ratio of the effect of the noise to the parents is roughly approximate or smaller by an order of magnitude.
For the nonadditive case, decompose the variance using the law of total variance,
Var
X Var [x U) U + Var E[f pas U) U..
Similarly, we choose the functional equation f such that f satisfies
Var E[ pa. U) U]
0.05 S E Var [f pa: U) U']
< 0.5
For all graphs, U, jid A/ 1), and choose of such that X; U; if X, is root node, i.e. f is the identity function. Lastly, we normalize every node X; such that Var (X,) R 1. For the sake of clarity, we omit all normalizing terms in the formulas and omit functional equations for root nodes below.
Results. For these 4 graph structures (chain, triangle, diamond, and Y), in Table 6 we provide the performance of all evaluated models for observational, interventional, and counterfactual queries, averaged over 10 separate initializations
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DCM
ANM
VACA
CAREFL
SCM
Metric
(x10-²)
(x10-3)
(x10
(x10-
Obs. MMD
0.27=2.11
0.19:10277
1.63:0042
4.25=1.12
NLIN
MMD
1.71
0.70
10.10+11881
8.63
Chain
MSE
0.16
2,43+:2449
25.99-16.47
19.02.4..01
Obs. MMD
0.12
1.51=
1.53=10.41
3.48=1.06
NADD
MMD
2.851:045
7.34
10,42:2.77
11.02 1.32
MSE
7..84
88.56=2.633
98.82:1.16
105.80=11.04
Obs. MMD
0.16##0111
0.12=0.07
3.12+00.3
4.64=1.03
NADD NLIN
Int. MMD
1.50
3.28=0.79
18,43#1.72
7.08=0.82
Triangle
CF. MSE
0.22
9.80 1.70
178.69-1.6.45
41.55
Obs. MMD
0.12
0.51
2.42-10.48
5.12.11.10
Int. MMD
2.81=1021
0.54=
11.09=00.85
4.17=00.4
CF MSE
26.28:16668
17.25
173.67#16.28
121.99:+31.34
Obs. MMD
0.11=00.05
0.14+00.08
2.29++0.69
6.82=+0855
NLIN
Int. MMD
0.23
1,40=#0144
9.50+
14.97=1.229
CF MSE
0.28:
1,22++027
28.79++4.02
5..3
Obs. MMD
0.21
1.00
1.51
3.39
NADD
MMD
1.54
5.62.10.1
5.37-10.72
6.51
CF MSE
33.45:22.06
47.55=2.56
60.67+3.22
52.41 =6669
Obs. MMD
0.22
0.13
2.77::0.5
8.28=1.29
NLIN
Int. MMD
3.21
0.36+
25.30+11.99
18.23
Dianond
CR MSE
14.74+66.09
32.02+:37.74
138.65 12.33
607.62 1221
Obs. MMD
0.25
0.28
2.361:0.4
5.50=
NADD
Int. MMD
0.23
4.40
12.54
16.34=1.72
CF MSE
1.36
8.58
57.40
24.61
Table 6: stanaad deviation of observational, interventional, and counterfactual queries of four different SCMs in nonlinear and nonadditive settings over 10 random initializations of the model and training data. The values are multiplied by 100 for clarity.
of models and training data, with the lowest value in each row bolded. The values are multiplied by 100 for clarity. Figure 2 we show the box plots for the same of experiments.
The results here are similar to those observed with the larger graph structures in Table DCM has the lowest error in 7 out 12 the nonlinear settings, with the correctly specified ANM having the lowest error in the remaining 5. Furthermore, DCM and ANM both typically have a lower standard deviation compared to the other competing methods, For the nonadditive settings, DCM demonstrates the lowest values for all 12 causal queries.
D.3 Semi-Synthetic Experiments
To further evaluate the effectiveness of DCM, we explore semi-synthetic experiment based on the Sachs dataset (Sachs et al. 2005 We use the real world graph from comprised of 11 nodes !T The graph represents an
Unttps: : /www///ww/k//////////// htt.l//ww..ommmmppppttttrr//////
screte- sma screte
#sachs
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intricate network of signaling pathways within human T cells. The 11 nodes within this graph each correspond to of the phosphorylated proteins or phospholipids that were examined in their study.
For our experiment, we sample data in a semi-synthetic manner. For the root nodes, we sample from the empirical marginal distribution. For non-root nodes, since the ground truth structural equations are unknown, we use random neural network the structural equation, as was done in Section [5] We report the performances in Table 7 We see that the performance improvements with our DCM approach corroborate our prior findings.
DCM
ANM
VACA
CAREFL
SCM
Metric
(x10-3)
(x10-3)
(x10-3)
(x10
Obs. MMD
0.31
0.21
0.53
7.30:10.95
NLIN
Int. MMD
1.25=#0.11
1.37+0.12 12
2.03++0.36
5.77+0085
Sachs
CF MSB
0.72=00.19
4.72+1.71
17.71+9923
9.59+
Obs. MMD
0.18=0.07
0.18+0.06
0.39+00224
1.10
NADD
Int. MMD
0.26
188110.51
2.21
5.2911..0
CF. MSE
1.99
4.6815.77
7.30.11157
8.3018.34
Table 7: Mean=standard deviation of observational, interventional, and counterfactual queries of the Sachs SCM in nonlinear (NLIN) and nonadditive (NADD) settings over 10 random initializations of the model and training data. The values are multiplied by 100 for clarity.
E Real Data Experiment II
In order to evaluate our DCM approach, we assess its performance in computing the Individual Treatment Effect (ITE), which is defined as D; := (1) Y;(0) where (0))) E R is the potential outcome of unit when is assigned to the control group, and Y1(1) is the potential outcome when i is assigned to the treatment group. Note that we do not observe D; for any unit i as for each unit in the training data set, we observe either the outcome under control or the outcome under treatment, but never both.
Now, in non-simulated data, it is impossible to know the counterfactual ground-truth as this event did not happen. However, averaging the ITE over all individuals, i.e., (1) Y Y()),, provides with an estimate of the average treatment effect (ATE), for which approach, we first estimate the ITE for each sample by estimating each corresponding counterfactual outcome. We then average these to obtain the ATE, which we can then compare with the ground-trutt -tuth ATE. Since our techniques are designed for constructing unit-level counterfactuals, and not ATE directly, the goal here is not to compare against other ATE estimation approaches, but rather demonstrate that the individual treatment effects computed through our counterfactuals are reasonably accurate.
have real-world datasets with known ground-truth. Therefore, to evaluate our
The following experiments are based on popular datasets. Each experiment was repeated 10 times using the same model hyperparameters for each problem, without fine-tuning to the specific problem. For each dataset, we use the ground-truth graph structure from the literature and assign a DCM model each non-root node. For root nodes, we can directly use the empirical distribution without the need to fit a particular model.
Infant Health and Development Program Dataset. The dataset aims at predicting the effect of specialized childcare on cognitive test scores of infants (Hill 201I). The goal here is to see if treatment (specialized childcare) improves the cognitive abilities of infants compared to infants that did not receive the treatment. The average treatment effect here is the difference between the expected cognitive test score under do(specialized child care and expected cognitive test score under do(specialized child care 0). The ground-truth ATE (on cognitive test score) associated with this dataset is 4.021.
Lalonde Dataset. The dataset contains different demographic variables (e.g., gender, age, education etc.) with the goal if training programs increase earnings Londe 1986). The average treatment effect here is the difference
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Dataset
Algorithm
ATE (computed from ITE))
Relative Absolute Error (%)
IHDP
DCM
4.013
0.199%
IHDP
ANM
3.957
1.592%
Lalonde
DCM
1579.59
3.672%
Lalonde
ANM
1516.20
7.538%
Table 8: ATE estimation through computing unit-level counterfactuals. The ground-truth ATE provided with the IHDP and Lalonde datasets are 4.021 and 1639.80, respectively.
between the expected earnings under do(training program 1) and expected earnings under do(training program 0). The ground-truth ATE (on carnings) associated with this dataset is 1639.80.
Results. The results are summarized in Table g As noted above, the focus of this paper is computing unit-level counterfactuals. The fact that the estimate of the ATE computed from via DCM approach matches the ground-trut truth ATE well can be seen as a sign that the unit-level counterfactuals were computed accurately. For baseline, we also show the results where replace DCM approach with additive noise model (ANM) approach for modeling the SCMs.
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Nonlinear Equations
Observational MMD
0.0%
0.05
6.00
0.06
0.00
0.64
0.04
0.02
0.02
0.00
0.00
0.00
0.00
Interventional MMD
0.12-
0.20
0.25
0.20
MMD
0.35
0.10
0.073
0.30
0.04
0.05
0.0 0.253
0.02
0.023
0.05
0.00
0.00
Counterfactual MSE
16
0.3
M50 15
0.2
0.1
0.1
DCN
BCN
Chain
Triangle
Diamond
Structural Causal Model
Nonadditive Equations
Observational MMD
0.07
0.05
0.05
MMD
0.04
0.04
0.03
0.01
0.03
0.00
0.00
0.00
0.00
Interventional MMD
0.150
6..2
0.37
0.325
0.36
MMD
0.08
0.35
0.075
0.08
0.24
0.3
0.25
0.090
0.20
0.23
0.000
0.00
0.30
0.09
Counterfactual MSE
1.25
2.0
0.5
0.8
1.5
MST
5.75
0.4
5.4:
1.0
65
0.2
6.2
DCC
AMM
VACA
CAREF!
no
DCM
ANR
VADA OREE
n.o
DCM
ANR
WICA CAREFE
DCM
J0NN
WACA CAREFE
Chain
Triangle
Diamond
Structural Causal Model
Figure Top: Nonlinear setting (NLIN), Bottom: Nonadditive setting (NADD). Box plots of observational, inter- ventional, and counterfactual queries of four different SCMs over 10 random initializations of the model and training data.
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11
U3
Us
UT
Ug
X3
X
X
U2
U4
U6
Us
U10
Figure 3: Ladder graph used in Section
X1
X₂
X3
X
X5
X6
X7
X
10
Figure 4: Example of random graph used in Section
with exogenous noise nodes omitted for clarity.
U1
U2
U1
X1
U1
U2
U3
X1
X₂
X1
X
U2
2
U
X
U3
2
X
X
X
X
U₂
U4
X₂
Ua
Figure 6: Triangle graph.
Figure 5: Chain graph.
Figure 7: Diamond graph.
Figure 8: Y graph.
Figure 9: Causal graphs used in our experiments in Appendix D.2
31


Reviews:
[{"review_id": "5P7EIrQBPQ", "reviewer": "Reviewer_Sktb", "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:\n* The positioning of the study and contributions are clearly explained along with many references.\n* The effectiveness of the proposed method has been verified by both artificial and real data experiments, and is sufficiently convincing.\n* The proposed method is natural and generic and should be of interest to the machine learning community.\n\nWeakness:\n* It is not clear whether the theoretical analysis in Section 4 is applicable to the proposed diffusion-based approach in Section 3.\nFor example, there seems to be no discussion on whether assumptions 2 and 3 of Theorem 1 are valid in the diffusion model.\nAssumption 1 has also only been experimentally verified, and there is no theoretical evidence that the diffusion model satisfies this assumption.\n* I could not discern the logical flow that led to the proposal in this paper.\nIn particular,\nin dealing with structural causal models,\nI would like to know why the diffusion model was chosen among various encoder-decoder models that are applicable,\nand what challenges it addresses that other models cannot.\nSince the theoretical analysis part of Section 4 can be applied to arbitrary encoder-decoder models,\nI do not believe this supports the usefulness of the diffusion model.\nThe fact that each node can be trained in parallel is also not limited to diffusion models.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a method for learning and inference of structural causal models based on the diffusion models.\nThe proposed algorithm for training works given the causal graph and observational data.\nTheir trained models are able to answer causal queries including observational, interventional,\nand counterfactual queries.\nEmpirical evaluations demonstrate that their proposed methods superior to state-of-the-art methods for causal queries.\nTheir contributions also include a framework for theoretical analysis of encoder-decoder-based structural causal models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "If the authors can address any of the points listed as weakness above,\nI would appreciate it.\nHowever,\nI am not strongly opposed to accepting this paper as it is.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1HkAZRfmIM", "reviewer": "Reviewer_A998", "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": "**Strengths**:\n- good clarity of writing, especially after Section 1\n- ample theoretical support along with concise proof-of-concept experiments\n- fairly intuitive approach but with a more expressive class of models than previous papers\n\n**Weaknesses**:\n- Section 1 is less consistently well-written than the rest of the paper", "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 diffusion-based causal models (DCMs), which use diffusion models to approximate the functions in a structural causal model, given a DAG and observational data. The intuition is similar to the classic approach of using regression methods to approximate these functions, but instead using more expressive and flexible diffusion models based on neural networks. DCMs allow answering interventional and counterfactual queries by constructing a Markov chain of samples from post-intervention distributions. The paper also offers bounds on the error of counterfactual estimates in more general encoder-decoder approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(2. is critical, the others are suggested)\n1. Go carefully through the intro again---it seemed to have more mistakes and less clear/polished writing than other parts of the paper (though it was more clear to me the second time through, after understanding more about diffusion models). For example:\n    - \"... between a Gaussian and a density.\" seems to be missing some words\n    - \"... these are first error bounds...\" seems to be missing \"the\"\n    - \"In contrast to our work, while Javaloy et al. (2023) focuses on modeling the whole causal graph as one model.\" Is the word \"while\" unnecessary, or is some other part of the sentence missing?\n2. \"Note that causal sufficiency is the minimal necessary set of assumptions for causal reasoning from observational data alone.\" Isn't this just one assumption and not a set? More importantly, this claim needs more explanation, context, or justification. Why doesn't something like latent causal discovery count as causal reasoning? \n3. \"For example, our DCM approach trains approximately seven times faster than CAREFL and nine times faster than VACA.\" This reads like a very general claim, but it's actually only supported by a seed on a single ground truth. Weaken the claim or add more support.", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "BG3qAkFGN1", "reviewer": "Reviewer_4cS6", "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": "strengths: the proposed method might have good potential in settings where sampling from the dataset is difficult, e.g., when the node of the causal graph is an image or text. The authors attempted to provide theoretical guarantees of the proposed methods under some relatively strong assumptions.\n\nWeakness: The paper could also benefit from improved writing in the introduction to clarify the contribution and objective of the paper. the proposed method relies on strong assumptions about the underlying data generating distributions which might be unrealistic when the data type is complex (e.g., text/image). For simpler setups, e.g., the nodes of the causal graph are of numerical values, it is unclear whether the proposed method will outperform semi-parametric estimation methods. When the data generating process is complex, it is unclear how the causal DAG could be provided/constructed in the first place. A better example is needed. The numerical results on the real-world data seem a bit marginal.\n\nOverall, I see the real-world potential use case of the proposed methods. However, this is hindered by the strong assumption made by the methods. A strong real-world case study will be needed for the paper to be publishable. I am happy to update the score if the authors can either 1) update their numerical results with better benchmarks and a stronger case study to support their contribution claim in the introduction, or 2) revise the scope of the contribution of the paper to acknowledge the limitation of the proposed methods at the beginning of the paper. The idea of using diffusion models to sample from DAG might be an idea that is of interest to a broader audience.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides the first attempt to use diffusion models in answering causal queries. The proposed method takes a causal graph as input, models each node in the causal graph as a diffusion model and learns how to sample from the data generating distribution by using diffusion models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "writing: \n1. please formally introduce the problem of causal queries in the introduction. In the preliminaries, instead of explaining counterfactual queries in words, please provide a mathematical definition: do you mean the expected outcome here?\n2. the first sentence of the second paragraph of the introduction is very confusing: \"we focus on approximating SCM given observational data and the underlying causal DAG\" This sentence sounds like you are learning the underlying DAG from data, but you are not. It is better to say along the lines of given a causal DAG, we focus on approximating the data generating distribution with diffusion models\n3. the fourth to the last row of page 1 \"which may be interpreted as a deterministic autoencoder model with latent variables\": this is a causal paper, what do you mean by latent variables here? do you mean unmeasured confounding? how does this relate to the assumption of causal sufficiency that you subsequently made in the paper?\n4. khemakhen et al. (2021) \"their approach can be extended to more general DAGs\": the proposed method scales exponentially in the number of variables in the DAG. This approach in general is not scalable and this is an overstatement.\n5. in the discussion of assumption 2 \"assumption 2 is always satisfied under additive noise model\": is this true? f here is independent of U, so it seems that it would not satisfy the second half of the assumption where you assume that f is strictly increasing in U. \"e.g., distinguishing between X:= X_pa + U and X:= X_pa - U\": does this example contradict with the claim that assumption 2 is always satisfied under additive noise model? The two U's here can be thought of as having flipped distribution.\n\nassumptions:\n1. in the preliminaries end of the second paragraph \"every SCM M entails a unique joint observational distribution satisfying the causal Markov assumption...\" this seems to imply that the causal DAG is acyclic. Can you verify this and make it more explicit?\n2. the Assumption 2 in Theorem 1 is too strong: I understand for diffusion models to work on this problem, you probably would require some form of assumption 2. However, I do not see how this assumption is easily satisfied in practice. Can the authors comment on the comparison to semi-parametric estimation methods which would not require this assumption? So long that I have a target parameter defined, I can estimate the data generating distribution nonparametrically (here we can also leverage the structure of the causal graph), perform a plug-in estimation, and subsequently perform a debiasing step to get asymptotically desired behavior.  \n3. In theorem 2, assumptions are listed without interpretation. What does it mean in practice for Jf_x_pa to be positive definite?\n\nexperiments:\n1. ANM does not require the monotone assumption, can you verify whether your real-world data satisfies the monotone assumption listed in Assumption 2?\n2. Can you provide some robustness check of your method when assumption 2 does not hold?\n3. From the description of the real-world experiments, it is unclear to me what the format of the input data is and what the causal graph is. Is the graph simply CG->HG? The real-world experiment is not very convincing that this method can be useful in practice when compared with simpler benchmarks. A better case study is needed to illustrate the real-world use case of the proposed method.\n4. Can the authors compare the proposed methods to statistical estimation methods? ANM is more commonly used as a discovery method than an estimation method. The other two benchmarks are not specialized in inference.", "broader_impact_concerns": "no concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviews were detailed and there was significant discussion between the reviewers and the authors. There are no further comments that came up in the discussion and all reviewers are supportive of the paper being accepted by TMLR. The authors should update the manuscript to remove the highlighting on the amendments and the paper should be ready for publication.
Decision: Accept (poster)