Stochastic Neural Networks for Causal Inference with Missing Confounders
Authors: Yaxin Fang
Faming Liang
Venue: ICLR 2026 Poster (2026)
Keywords: Adaptive Stochastic Gradient MCMC
Causal Inference
Latent Variable Model
Missing Confounder
Stochastic Neural Network
Primary area: causal reasoning

Abstract: Unmeasured confounding is a fundamental obstacle to causal inference from observational data. Latent-variable methods address this challenge by imputing unobserved confounders, yet many lack explicit model-based identification guarantees and are difficult to extend to richer causal structures. We propose Confounder Imputation with Stochastic Neural Networks (CI-StoNet), which parameterizes the conditional structure of a causal directed acyclic graph using a stochastic neural network and imputes latent confounders via adaptive stochastic-gradient Hamiltonian Monte Carlo. Under SUTVA and overlap, and assuming that the structural components of the data-generating process are well approximated by a capacity-controlled sparse deep neural network class, we establish model identification and consistent estimation of the mean potential outcome under a fixed intervention within this class. Although the latent confounder is identifiable only up to reparameterizations that preserve the joint treatment–outcome distribution, the causal estimand is invariant across this observationally equivalent class. We further characterize the effect of overlap on estimation accuracy. Empirical results on simulated and benchmark datasets demonstrate accurate performance, and the framework extends naturally to proxy-variable and multiple-cause settings with overlap diagnostics and bootstrap-based uncertainty quantification.

Published as a conference paper at ICLR 2026
STOCHASTIC NEURAL NETWORKS FOR CAUSAL IN- FERENCE WITH MISSING CONFOUNDERS
Yaxin Fang
Department of Anesthesiology, Perioperative and Pain Medicine
Stanford University Palo Alto, CA 94305, USA yxfang@stanford.cem
Faming Liang
Department of Statistics Purdue University
West. Lafayette, IN 47907, USA
fmliang@purdue.cm edu
ABSTRACT
Unmeasured confounding is a fundamental obstacle to causal inference from observational data. satent-variable methods address this challenge by imputing unobserved confounders, yet many lack explicit model-based identification guarantees and difficult to extend to richer causal struc- tures. We propose Confounder Imputation with Stochastic Neural Networks (CI-StoNet), which parameterizes the conditional structure of causal di- rected acyclic graph using a stochastic neural network and imputes latent confounders adaptive stochastic gradient Hamiltonian Monte Carlo. Un- der SUTVA and overlap, and assuming that the structural components of the data-generating process are well approximated by a capacity-controlle sparse deep neural network class, we establish model identification and con- sistent estimation of the mean potential outcome under a fixed intervention within this class. Although the latent confounder is identifiable only up distribu-
to reparamcterizations that preserve the joint treatment-outcome tion, the causal estimand is invariant across this observationally equivalent class. We further characterize the effect of overlap on estimation accuracy. Empirical results simulated and benchmark datasets demonstrate accu- rate performance and the framework exlends naturally to proxy-variable and multiple-causs settings with overlap diagnostics and bootstrap- based uncertainty quantification.
INTRODUCTION
Causal inference from observational studies is topic of significant interest in fields such as genetics, economics, and social science. Under the potential outcome framework (Rubin 1974), fundamental condition for identifying causal effects is the strong ignorability condition
|Roeenaum & Rubin 1983
A IL (a : a E A) Z.
(1)
where A denotes the treatment variable taking values in the space A, Y(-) denotes the outcome function, and denotes confounders. A confounder refers to a variable that influences both the treatment and the outcome. The strong ignorability assumption requires that all confounders be observed. In observational studies, this condition is seldom fully met, thereby introducing the risk of substantial bias in causal effect estimation.
"The code of the experiments is available at: https://github. wtt.//ccc.///////c
Baochatte-Mearal--eetooke------- hh Missing- Confounders

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One strategy to address the issue of missing confounders is to model them latent variables. Wang & Blei (018) proposed using a latent factor model to obtain a latent representation for multipll causes, enabling the capture of multiple-cause confounders under the assumption that no single-cause confounder exists. Kallus et al. (2018) tackled this problem under a proxy variable setting by leveraging the low-rank components of the proxy variables, obtained through matrix factorization, as an approximation to the true confounders. [ouizos et al. (2017) also addressed the issue with proxy variables and introduced the causal effect variational autoencoder (CEVAE) to infer missing confounders from the obscrvational distribution of the proxy, treatment, and outcome. These works represent significant advancements in causal inference using observational data; however, they have nolable limitations. For instance, Imai & Jiang (2019) pointed out that Wang & Blei (2018) essentially models the substitute conlounder as a deterministic function of treatments, leading it to converge to function of the observed treatments rather than the true confounder. [Kaluu et al. (2018) focuses primarily on the linear regression setting, limiting its applicability to nonlinear models unless many proxies are available for a small number of latent variables. Rissanen & Marttinen 2021 examined the consistency of the causal effect estimator in Louizos ct al. (2017) and showed that it fails to correctly estimate cause effects when the latent variable is misspecified or the data distribution is overly complex.
In this paper, we propose a latent variable imputation approach to address the issue of missing confounders. This new approach is built on the stochastic neural network (StoNet) (Sun & Liang] 2022 Liang et al. 2022) and sparse deep learning theory (Sun et al. 2022). effectively overcoming the limitations of existing approaches.
The core idea is to encode the causal DAG's Markov factorization in StoNet and to impute latent confounders through the resulting conditional distributions. Importantly, our theoreti- cal guarantee is model-based: under SUTVA and overlap, and assuming the treatment and outcome mechanisms lie a function class that can be well approximated by sparse DNNs, the causal functional (a)) is uniquely determined and is consistently estimable.
The StoNet is trained using an adaptive stochastic gradient MCMC algorithm (Liang et al. 2022 Deng et al. 2019), which allows for the sinultaneous imputation of missing confounders and estimation of sparse StoNet parameters. We refer to the proposed approach as Confounder Imputation with Stochastic Neural Networks (CI-StoNet). In summary, it offers the following advantages in addressing the missing confounder issue:
(i) Accurate Causal Effect Estimation: This property is supported by StoNet's inherent ability to handle missing data, the consistency of sparse deep learning, and the convergence guarantee offered by the adaptive stochastic gradient MCMC algorithm.
(ii) Complex nonlinear modeling. CI-StoNet inhorits the universal-approxinattie prop- erty of deep neural networks (DNNs), enabling effective modeling of complex nonlinear relationships across diverse applications.
(ifi) Structural flexibility. The Markovian architecture of CI-StoNet provides structural flexibility for representing diverse dependency patterns in causal DAGs. It supports localized updates to each DNN module, promoting modular design and easy adaptation to varying causal relationships.
2
CI-STONET FOR MISSING CONFOUNDERS
2.1 THE CI-STONET APPROACH
This section introduces the Cl-StoNet approach or, more generally, deep learning framework for performing causal inference in presence of missing confounders.
Consider the scenario of simple confounding, as depicted by Figure ! which involves treatment A E (aI..... am }: missing/ /latent confounders Z, and an outcome Y. The corresponding model is given by
Y
A gi (Z, ea);
(2) Figure 1: simple confounding
Y = 92(Z, A) + Ey:
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where 91(-) and 92(+) are unknown functions that can be nonlinear and highly complex, and ea and Ey are random errors. In this paper, we assume ey N(0 o(0,o)44,) where dy denotes the dimension of Y. There is flexibility in specifying the distribution of eg. If A is continuous, then 6a can be assumed to follow a Gaussian distribution or any other continuous distribution. If A is mixed, the distribution of each component of ea can be specified accordingly. This scenario has included multiple causes considered in Wang & Blei (2018), where a; multi-dimenniionl vector, as special case. Since Z missing, impute it from the conditional distribution:
< (Z|A, Y) π ((Z) (A) I A) T (ZIA)) A),
(3)
where TT-) denotes distribution or conditional distribution in the appropriate context.
The latter part of Eq. (3) suggests that, mathenatically, A, Z, and Y can be interpreted as the exoge- nous imput, latent state and output of a stochastic model. Motivated by this view, we propose to perform the imputation using CI-StoNet (see Figure 2 for its structure), for- mulated
Treatment
2,
Outcome Y
Treatment
Z /1 (1 +es
(4) Figure 2: Diagram of CI-StoNct under simple confound- ing, where white rectangles represent variables from observed data; light-grey rounded-reclangles represent latent variable to impute; and dark-grey reclangles rep- resent neural network modules to learn respective ditional distributions.
= (22 Z, 02) + evy
where (1) and (2(+)) are two neu- ral network functions, parameter-
ized by 01 and 02, respectively;
e₂ ~ (O, a ) and ey as de-
fined in (2). The two neural networks are interconnected through the latent variable Z. Additionally, we impose the following assumptions on the models ( and (4):
Assumption 1. (i) ea CaJlZ, CyJl(Z, Z A); (ii) there exist sparse DNNs n1(+) and u2(-) such that (4) holds, e2 ~ N(O, of ey o0,,o Tā))), ea and e, A;
Part (i) of Assumption 1 lensures that strong ignorability (1) holds. Part (ii) specifies the working model class by restricting the treatment and outcome mechanisms to sparse DNN family (with Z random). This structural restriction enables tractable theoretical analysis and defines the pseudo-true parameter 0* that minimizes the KT divergence from the true data-generating law. See further details in 2.2.1
Notably, the functional expression in 44 docs not imply causal mechanism A + Z. For example, rain (Z) causes a wetland (A), but a welland does not cause rain. Similarly, Z cannot be interpreted as a mediator due to the nonexistence of causal mechanism A Z, although
{ has mathematical structure similar to mediation models. For the time being, and the
we assune that there no mediator in the causal pathway between the treatment outcome Y, thereby ruling out any potential misinterpretation for the role of Z. However, if mediator does exist, issues related the total causal effect estimation and the interpretation of Z will be addressed at the end of Section 2.2
Under the missing data framework, the CI-StoNet can be trainod by solving the following equation, which represents Bayesian version of Fisher's identity (Song et al. 2020
Vo log n (0AAY)
Vo log क (0 Z, A, Y ()A,
(Z)A, where 0 [0,, 02). Z missing, π (0) A.Y) x ( )T {
Z, A, 02),
T (2 A. (Y A,0), and π (01)) and =02)) denote the prior distribu- tions imposed on 0, and 02, respectively. In this paper, we assume that the components of 0 are a priorí independent and are subject to the following mixture Gaussian prior Sun ot al. )022):
T (2 A, A,A,Y,
Kn
T (0) = II (1 \m)0(0//00) + (0.(00/))),
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where An is the mixture proportion, Kn is the total mumber of connections in the StoNet (i.e., the dimension of 0), o(-) represents the density function of the standard normal distribution, and dg and of are the standard deviations of the two Gaussian components, respectively. The identity
further suggests that the target equation
Ve log (0 Y) = O,
(7)
can be solved using an adaptive stochastic gradient MCMC algorithm, which iteratively alternates between latent variable imputation and parameter updates. In this paper, employ the adaptive stochastic gradient Hamiltonian Monte Carlo (SGHMC) (Liang et al.
2022), given in Algorithm to solve equation (
Algorithm 1: Adaptive SGHMC
0. Set the prior hyperparameters: An, do, and d1
1. (Latent variable imputation) Simulate from xt AZ Y. 0) via Hamiltonian Monte Carlo updates:
v(k+1) (1 (1-e411))v*) ex+1 Vz log 14 ca+s " A, ))
20 (k+1)
(++1) zz + (k)
where el++1) ~ (0, Ld.), de is the dimension of Z, and Ea+1 is the learning rate.
2. (Parameter update) Given Z (1+11,, update 0, and O2 separately:
0 (k+11) = (4()) + 7h+1 Vo, log 2 (++1) A, x(*)) + 7k+1 Ve, log (0(x)) x (k))
0 (k+1) = (/4)) + 7 Ve, log Y (H-1)
A,
+ 7h+1 Ve, log क (
In model (4) both 02 and By are scalar. They can be treatod hyperparameters specify in simulations, while having minimal impact on the downstream inference. Notably, Ø, is essentially non-identifiable in model 4 due to the universal approximation property of neural networks. In the inference stage, see equation g9. we provide Bayesian estimator for of to facilitate imputation of missing confounders. Specifically, we impose an inverse gamma prior of ~ InyGamma(a, B), leading to the Bayesian estimator:
3 3+
E}=1(e (2) 01)))))))²
0²
(8)
3 +
where we set a 3 for flat prior, and Aj denotes the value of A in sample j. In simulations, its value can also be updated as in (5) along with iterations, while having minimal impact on the performance of the algorithm.
To enable causal inference, we introduce the following additional assumptions, which are standard conditions for causal effect identification:
A Assumption 2.
1. Stable unit. treatment. value assumption (SUTVA): the potential outcome of one subject are independent of the assigned treatment of another subject; that is, there is no interference between subjects and there is only single version of each assigned treatment.
2. Overlap: The substitute confounder Z satisfies the overlop condition: p(A A|Z) for all sets A with positive measure, i.e., p(A) > 0.
Under Assumptions 1 and 2 the causal effect can be estimated in the following procedure:
Causal effect estimation. After Algorithm converges, with learned parameters 0' (0), 02), for each observation draw M samples (z()) ) from m(s a 0)) The expected M
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outcome (aY(a) [00} a2( a: 02) T( (i) dz is then approximated by the Monte Carlo average
((Y(a)) n.M
U2(2) a,0, 07)
justification for the estimator is given in Appendix can then be derived accordingly.
A5.3
The treatment effect estimator
2.2 THEORETICAL GUARANTEES
2.2.1 MODEL-BASED IDENTIFIABILTTY
Following Rissanen & Marttinen 2021], , we distinguish nonparametric identifiability from model-based identifiabilit.. Nonparametric identifiability concerns whether the causal effect can be recovered solcly from the observational distribution, whereas model-based identifiability asks whether the causal effect is unique within restricted model class. Our results pertain to the latter. Concretely, we restrict the latent-variable model class by:
Assumption 3. (Counfounding mechanism) The structural conditional mean functions ma(=)= = E[A] Z =] and my (a, z) E|Y|A= a. Z z] belong to function class F on a bounded domain such that there erists a sequence of sporse DNNs PA,. and jy,n satisfying Assumplion A10 with
IA - mall + ll - my S wn
where Wn some sequence converying to as n -
This condition holds for many function classes F, see detailed discussion in Appendix A5.4 Under such restriction, let {Po 0 E On) denote the family of observed-data laws induced by CI-StoNet under the working causal DAG and the sparse DNN specification (Assumption (ii). For the target functional (a)(a) = Ep, [a)]; we say ve(a) is model-identifiable with respect to Pe if, whenever two parameter values 0, 0 induce the same observed-data distribution (e.g. pe (A,A, in simple confounding or (A,Y, X) in proxy settings), it follows that vo(a) vigg (a) for all a.
In our setting, Z and 0 may be non-unique due to loss- ioss-ivvriant transformations, but the causal functional is invariant within each obsorvational equivalence class; thus, Theorem A1 parameter recovery up to such transformations) and Theorem [ consistency of (a)))) presented below, logether yield model-based identification and consistent estimation of (a)) within the specified family. When the true mechanisms fall outside the sparse DNN family, Theorem 2 quantifies the resulting misspecification bias.
2.2.2 THEORETICAL ANALYSIS
Let Po denote the true joint law (A, ZYY, Po be the distribution induced by the CI-StoNet with parameter 0. Define a pseudo-true StoNet parameter
[1o8 dPo
0
arg min KL(Po, Po) = arg min
dPo
a.Po
Let (PP) = my (a, = (z) be the true potential outcome, v/(P) B(Y = (2(a, z)Pp.- (z)dz the pseudo-true potential outcome, and ((P)) B(Y
TM ΣE EM (2( 0z). The error decomposition shows:
[P)(P)) (Pooll < - x + (Po) (
estimation error
misspecification enror
Theorem 1. (Estimation SSppooe Assumptions
2 and the conditions in Lemma A1
and Theorem A1 (stated in Supplement A5 hold. Then
[1 100--/(PP ( 0, as M + 0o and co.
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Remark 1. As shown in the proof of Theorem the consistency of the estimator arises from the existence of the true sparse StoNet as well as the consistency of 0 a. is importaat
to nole thot, due to the non-uniqueness of 6n discussed in Remark A1 the impuled latent confounders may differ from their true values. However, T(2 A, A, 0 still serves as a
consistent eslimalor (in terms of the densily function) of x==|A,0)) 0ï), up to loss-invarianl transformation of p,. Nevertheless, this does not affect the consistency of the estimator which is remarkable property.
Thcorem 2. /Misspecciication error) Fix a treatment level a. Let Po denote the true (complete-dtta law of (A, Z. Y), and let {Pa 0 E On) denote the family of (complete-dates laws induced by the CI-StoNel working model with the sporse-DNN archilecture at sample size n. Assume that the structural conditional mean functions mA and my = Ep, lY AA= a, Z = z] satisfy Assumption 3 and Assumption A10 with approrimation rale wa, and that the sparse-DNi DNN architecture satisfies Assumplion A13. Furthermore, assume the model outcome regression is uniformly bounded at any fixed treatment a: sup, lu2(a, z; 0=) S Cua.
Ep, [ |
Define the causal functional (PP) :- Ep(y In particular,
(PP)) = mmy myy(a (a, 2) pr. (2)r(2) dz,
((Po) = 12(a, 2: (z) (z)dz.
Then there exist constonts C1, C2 < 00 (independent of n) such that
KI(Po, Po.) S C1 w (Pb)(P)) - s/( Po. C2 Cn
A finite-sample error analysis for both the simple confounding case (Theorem A2 and the basic proxy case (Theorem A3 is provided in Supplement A5.55 The analysis shows how the estimation error bound of potential outcome worsens when there is bad overlap P(A|Z).
Notably, by approximating the treatment and outcome mechanisms using sparse DNNs, CI- StoNet imposes relatively mild structural restrictions the confounding process. In contrast to WWann & Blei (2018), it does not rely on a "no single-cause confounder" assumption and thus accommodates both multi-cause and single-cause confounding within unified framework. Moreover, unlike the variational autoencoder approach of Louizos et al. (2017), which may yield latent representations without consistency guarantees when large neural networks are used, Ne- leverages sparse deep learning theory and Bayesian regularization to ensure parameter estimation consistency within the specified model class, thereby providing thcoretically grounded causal effect estimation relative to the pscudo-true parameter.
Additionally, we note that the latent variable imputed in step (i) of Algorithm cannot be used for causal elfect estimalion, as it may contain information related to colliders. Figure 3(a) illustrates this concept, where the collider variable is influenced by both and Y. In step (i), we impute Z conditioned
both A and Y. If collider variable exists, the imputed latent variable may introduce spurious associations between A and Y, potentially biasing the causal effect estimation. To mitigate this issue, specifically impute the latent variables from (Z|AA,0) 0)), ensuring that any collider-related information is excluded from the analysis.
(a) Collider
(b) Mediator
In Section 2.1 we assumed the absence of media- tors to enable a clear interpretation of Z as la- tent confounder. However, if a mediator M does exist, illustrated in Figure 3(),, the imputed latent variable Z may inadvertently encapsulate information related to M. Mathematically, the conditional distribution can be expressed as:
A, <T A, 2) 2)= ()) (M A) dM,
Figure 3: Other examples of causal struc-
tures: (a) existence of colliders, repre- sented by C (b) existence of mediators, represented by M.
which indicates that, without observing M, its effect will be absorbed into Z, making them
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statistically indistinguishable within the CI-StoNet framework. In this case, the treatment effect can still be estimated based on (9) where acts as a latent adjustment variable that facilitates estimation of the total causal effect. Although this precludes pathway-specific interpretations, it does not invalidate estimation of the total causal effect. If, however, the mediator M is known from domain knowledge or experimental design and there is no unmeasured confounding between A and M, or between M and Y, then the front-door criterion (Pearl 2009) can be applied. In this case, M can be included as part of the latent confounder layer in the CI-StoNet to enable identification of the direct causal effect via front-door adjustment.
2.3 A SIMULATION STUDY
As concept-proof example, evaluated CI-StoNet using simulation study with a nonlinear tata-generating process for A and Y under both separable and non-separable confounding scenarios. We generate the latent confounders Zi...., Ze as independent standard Gaussian random variables, and then draw AI,..., Ay independently from the distribution, using inverse CDF:
p(a.lZ) expit({(2)a,)
1 expit(E(Z)a,) 1-15a.513 1=
expl((Z)a.) where expit(E(Z)a;) I+exp(((2)a.)> and {(Z) Ei=1 Bisina Ej= cos 2j
Exss I+expl - . We fi(A) 8 and f2(A) - Eiej a;a,, where 82 represent an element-wise square operation, and generate Y in two settings: (i) Sep- arable confounding. the treatment and confounder impact the outcome separately: Y = [A( - (A))(A) + E(Z) + +,, where 0n ~ U(-1,1) and e ~ (O. (ii) Non- separable confounding. there exists interaction between the treatment and confounder: == fAA - (()) + E(Z) + e, where ~ NO0, 1).
For each setting, the experiment was conducted on 10 simulated datasets, each comprising 1000 training samples, 500 validation samples, and 500 test samples. The marginal treatment effects were calculated using the set. Figure compares the and estimated marginal treatment effects across the 10 datasets. The plots show that most of the estimated marginal effects lie within half standard deviation of the true marginal effects, indicating that C1-StoNet is able to estimate the marginal effect of each treatment with small bias.
3 CAUSAL STONET FOR PROXY VARIABLES
In some applications, proxy variables for an unobserved confounder are available, though they may be noisy or only partially informative. Conditioning on such proxies can reduce, but not fully remove, confounding bias, making it natural to incorporate them as substitutes for the missing confounder. Kuroki & Pcarl (2014) established identification results using matrix adjustment or spectral methods under specific assumptions
measurement eтrоr. al. (2020) and Miao al. (2018)
The proximal causal inference framework of Tchelgen showed that, causal effects can be identified when two types of proxies treatment and outcome confounding proxies are observed. In contrast, Louizos et al. (2017) proposed variational autoencoder approach that leverages a basic proxy to learn latent confounder representation.
Consider the causal structure with basic proxy, depicted in Figure a). This causal structure suggests that Z can be imputed based on the following conditional distribution:
म (Z X) & (
A) x n Z X
Z, A).
(10)
The decomposition of the conditional distribution
10
further suggests the StoNet structure:
Z nn ( ez, A = n2 (2,(Z,00 0,0 Ca), Y 14 Z2 A, 0g) + ey:
(11)
where e, and e, are Gaussian random errors, while the form of e, can be determined according to the types of treatments. These random errors mutually independent and

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(a)
(b)
0OOD
Proxy
Af 2,49
Outcome Y
Treatment
Figure 4: (a) Causal DAG: without dependence on the proxy; (b) Diagram of CI-StoNet under the proxy setting: white rectangles represent variables from observed data; light-grey rounded-rectangles represent hidden neurons; lark-grey roctangles represent network modules to learn respective conditional distributions.
also independent of X. Figure 4(b) illustrates the corresponding CI-StoNet structure. Alternatively, we can consider the following StoNet model:
= (x +e.s
u2 (Z, 02) ea. Y = A, 03) + Ey:
(12)
where Es, e, and Ey are Gaussian random errors. Notably, the model [11 and the model 12) are asymplotically equivalent, even when A is binary vector. In the binary case, their equivalence is supported by the result that, as shown in Liang (0033) and Duda et al. (2001).
U2 Z, 02) converges to the probability function P(A 71A=12,0 02)
In this paper, we adopt the model [2) for computational simplicity. A gradient equation analogous to 7 can be constructed for the model. An adaptive SGHMC algorithm, similar to Algorithm can be employed for its solution. Let {z(" 1,2,.... M denote the samples simulated from <<2|,,, 07). Then the expected outcome function E(Y (a)) be estimated by the Monte Carlo average as
E(Y(a) n. ΣV
ps(z!) aa 0,)
(13)
3.1 NUMERICAL EXPERIMENTS
For simplicity, consider single binary treatment in experiments. CI-StoNet is compared with the following basclines:
(i) Designed for average treatment effect (ATE): double selection estimator (DSE) Belloni
et al. 2014), approximate residual balancing estimator (ARBE) (Athey et al. 2018). largeted maximum likelihood estimator (TMLE) (van der Laan & Rubin 2006), and deep oothoganal networks for unconfounded treatments (DONUT) (Htt & Feuerriegel 2021
(ii) Designed for beterogeneous treatment effect: X-learner (Künzel al. 2017). Drag- onnet et al. 2019), causal multi-task deep ensemble (CMDE) (Jiang et al. 2023
causal multi-task gaussian processos (CMGP (Alaa & van der Schaar 2017)), causal effect
variational autoencoder (CEVAE) [Luuizos et al. 2017), generative adversarial networks (GANITE) (Yoon et al. 2018), and counterlactual regression net. (CFRNet (Shalit et al. 2017)). For the baselines in part (ii), we use the code of Jiang et. al. (2023) at GitHub.
For performance evaluation, we consider two metrics: (i) estimation accuracy of ATE, which is measured by the mean absolute error (MAE) of the ATE estimates; and (ii) estimation accuracy of CATE, which is measured by precision in estimation of heterogeneous effect (PEHE).
3.1.1 SIMULATED EXAMPLES
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This example is designed to compare methods on problems with nonlinear treatment effect and non- linear outcome function. We generated 10 datasets using the procedure de- scribed in Section A1.2 with each dataset consist- ing of 2000 training sam- ples, 500 validation sam- ples, and 500 test sam- ples. Table shows that C1-StoNet provides accu- rate estimates for the het- erogencous treatment ef- fect and outperforms the baselines.
Table 1: Comparison of different methods for estimation of het- erogeneous treatment effects with proxy variables, where PEHE was computed over 10 datasets, 'In-sample PEHE' was computed with training and validation samples, and 'Out-of-sample PEHE was computed with test samples.
In-Sample PEHE
Out-of-Sample PEHE
CI-StoNet CMDE
0.3614(0.0328)
0.3731 (0.0350)
0.9019(0.0746)
0.9059 (0.0699)
CMGP
1.8823(0.0836)
2.2116 (0.1682)
CEVAE
0.6190(0.0350)
0.6246 (0.0384)
Ganite
1.2099(0.0558)
1.1797 (0.0499)
X-learner-HH RF
0.8308(0.0200)
1.4272 (0.0132)
X-learner-Bart
0.6489(0.0168)
0.6570 (0.0151)
CFRNet-Waas
1.7127
1.7258 (0.1667)
CFRN
2.0238(0.0537)
2.0250 (0.0582)
DragonNet
0.4217(0.0356)
0.4305 (0.0361)
3.1.2 BENCHMARK DATASETS
We evaluated CI-StoNet on some benchmark datasets, including the Twins dataset and 10 datasets from Atlantic Causal Inference Conference (ACIC) 2019 Data Challenge. The results reported in Section A1.3 indicate that CI-StoNet outperforms the baselines.
4 CONCLUSION
By integrating StoNets with adaptive stochastic gradient MCMC, this paper develops practical and thcoretically grounded framework for causal inference with missing confounders. CI-StoNet encodes the dependence structure implied by the causal DAG and estimates parameters using sparse deep learning with Bayesian regularization. Although the latent confounder is identifiable only up to loss-invariant transformations, the induced causal effect well-defined within the sparse StoNet class. The framework exlends naturally to settings with multiple causes and proxy variables.
Within this framework, the causal functional model-identified: observationally equivalent parameterizations yield the same causal effect. Moreover, the total error decomposes into statistical cstimation error and misspecification error relative to the pscudo-true parameter, allowing explicit characterization of convergence and model approximation.
Despite its advantages, this study has some limitations. First, the structure and parameter estimation of CI-StoNet rely on correct specification of the underlying causal DAG. For example, in settings with multiple treatments, an unrecognized mediator may be absorbed into the learned substitute confounder, potentially biasing causal estimates, since the model is defined through the joint distribution of (A, Y Z). If the mediator is properly identified, however, the CI-StoNet structure can be modified accordingly avoid this issue. Second, the current formulation of CI-StoNet does not provide principled, model-based uncertainty quantification for the causal effect estimator. Although the method does not inherently yield valid posterior intervals for the causal functional, we provide practical pess-processing procedure in Appendix A4 that constructs bootstrapped confidence intervals for the estimated causal effects. More formal uncertainty quantification could be achieved by adopting the original StoNet framework [Lingg et al. 2022), which enables inference through its fully Bayesian formulation.
Finally, the Markovian structure of CI-StoNet affords substantial flexibility for modeling a wide range of causal structures. Section A2 extends Cl-StoNet to two proxy-variable settings: (i) outcome depending on the proxy and (ii) treatment depending on the proxy. See that section for details.
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A
APPENDIX
A1
SUPPLEMENTARY EXAMPLES
A1.1 FIGURES FOR THE SIMULATION STUDY IN SECTION 2.3
THPPOR
12
(a) Separable confounding
(b) Non-separable confounding
Figure S1: CL-StoNet results for the simulation study, Points denote the true marginal effects and error bars represent standard error of the marginal effect estimator.
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A1.2 SIMULATED EXAMPLES
The simulated examples are used to compare the performance of different methods for problems with nonlinear trealment effect and nonlinear outcome function. We generated ten datasets using the following procedure, with each datasel comprising 2000 training samples, 500 validation samples, and 500 test samples.
1. Generate the confounder 2; (2 2,55) independently from N, 1).
2. Generate Yi, ".,1.".. 10,100 independently from (P,, 1) truncated in the interval [-10, 10], with μ4 = E Fi Set the proxy variable x; (Zi,....,, F1,00) with
Ti.j = Mir where æ and 2 are dependent through
V2
3. The propensity score p(=;)= [1 ( + ((2,,)) + ((2,))))) where S2,1 is the CDF of the beta distribution with shape parameters (2, 4), and & denotes the CDF of the standard normal distribution. This ensures that p(z,) E [0.25, 0.5], thereby providing sufficient overlap. Treatment A is hence generated from a Bernoulli distribution with the success probability p(zi). Resampling from the treatment and control groups has been performed for ensuring that the dataset contains balanced samples for treatment group and control group.
4. To simulate the outcome,
y, c(z,) + (T + m(z))) oye:
52g3
c(z,) = 1 2/4
+ 2245.
2 where n(x,) = f(zn)f(22) - (f and f(w) =+ That is, we set the Treatment effect T(2,) = = n222), which is homogeneous for different individuals. We generated the samples under the setting T 3, By 0.25, and E ~ 1).
A1.3 BENCHMARK DATASETS
We compare the performance of the proposed method some benchmark datasets, including the Twins dataset and 10 datasets from Atlantic Causal Inference Conference (ACIC) 2019 Data Challenge.
ACIC 2019 Datasets We first worked 10 ACIC 2019 datasets. This experiment locuses on comparing CI-StoNet with the baselines designed for ATE estimattion. The results summarized in Table SI which indicates that Cl-StoNet outperforms the baselines.
Table S1: ATE estimation across 10 ACTC 2019 datasets, where the number in the parentheses represents the standard deviation of the MAE, with additional benchmarks
Method
In-Sample
Out-of-Sample
CI-StoNet
0.0669 (0.0166)
0.0709 (0.0133)
CMDE
0.0802 (0.0166)
0.0877 (0.0246)
CMGP
0.1252 (0.0156)
0.1349 (0.0170)
CEVAE
0.0773 (0.0152)
0.0875 (0.0154)
GANITE
0.1622 (0.0390)
0.1747 (0.0425)
X-Learner-RF
0.1720 (0.0257)
0.1903(0.0253)
X-Leerner-BARE
0.0738 (0.0251)
0.0817(0.0248)
CFRNet-Wass
0.1024 (0.0241)
0.1099
CFR
0.1105 (0.0258)
0.1208(0.0246)
Twins Data.
We analyzed a real-world dataset of twin births from 1989 to 1991 in the United States. The treatment. variable is binary, with '1' denoting the heavier twin at birth.
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The dataset contains 46 variables that include clinical information and socioeconomic status of parents, and we regard them as proxy variables for latent confounders. The outcome variable is binary, with '1' indicating twin mortality within the first year. We regard each twin-pair's records as potential outcomes, allowing us to find the true ATE. After data pre-processing, we obtained dataset with 4,821 samples. In this final dataset, mortality rates for lighter and heavier twins are 16.9% and 14.42% respectively, resulting in true ATE of 2..8%.
We conducted the experiment in three-fold cross validation, where we partitioned the dataset into three subsets, trained the model using two subsets and estimated the ATE using the remaining one. Table S22 (left panel) reports the averaged ATE over three folds and the standard deviation of the average. CI-StoNet yields a more stable ATE estimate (in RMSE) compared to the baseline methods.
Table S2: Comparison of different methods in average treatment effect (ATE) estimation for Twins data, where the number in the parontheses represents the standard deviation the absolute of ATE, and RMSE denotes the rool mean squared error.
With confounder gestat10
Missing confounder gestat10
Methods
Absolute Error of ATE RMSE
Absolute Error ATE
RMSE
CI-StoNet
0.0099(0.0089)
0.0133
0.0135 (0.001)
0.0153
DSE
0.0157(0.0176)
0.0236
0.0211 (0.0193)
0.0286
ARBE
0.0152 (0.0201)
0.0252
0.0168 (2.0277)
0.0307
TMLE(Lasso)
0.0855 (0.0599)
0.1044
0.0932 0.991
0.1222
TMLE
0.1042 (0.0779)
0.1301
0.1238
0.1379
DONUT
0.0490 (0.0128)
0.0506
0.0490(0.0124)
0.0505
CMDE
0.0108(0.0905)
0.0911
0.0635
0.1106
CEVAE
0.0249
0.0249
0.0327(0.0633)
0.0712
Ganite
0.3519 (0.1533)
0.3838
0.4198(0.2278
0.4776
R-Fearner-RF
0.0056 (0.0257)
0.0252
0.0157
0.0301
X-learner-Bart
0.0194 (0.0192)
0.0273
0.0251(0.0312)
0.0400
CFRNet-Waa
0.0189 (0.0425)
0.0465
0.0211 (0.025)
0.0330
CFRNet-MMD
0.0439 (0.0146)
0.0463
0..619))
0.0639
Finally, to provide convincing evidence that the proposed method performs well when confounders are missing, we conducted an experiment where a significant confounder, gestat10 (gestational age), is intentionally omitted. In preprocessing the dataset, we followed Louizos ct al. (017) to focus on the same-sex twin pairs with birth weights less than 2 kg, and used the variable gestat.00 to generate "pseudo treatment assignments", Since gestat10 also an important factor for newborn mortality, it serves as a significant confounder. We removed gestat10 from the dataset. The results in Table S2 (right panel) show that CI-StoNet exhibits robust performance in presence of missing contounders. In this scenario, it outperforms all baselines in both the absolute error of ATE and RMSE, indicating the superiority of CI-StoNet gained from latent confounder imputation.
A2 EXTENSION TO OTHER CAUSAL STRUCTURES
The Markovian structure embodded in CI-StoNet provides it with great fiexibility to model wide range of causal structures. In this section, we extend CI-StoNet to handle other causal structures involving proxy variables. Specifically, we consider two scenarios: the outcome depending on the proxy, and the treatment depending on the proxy.
A2.1 OUTCOME DEPENDING ON PROXY
When outcome depends on proxy, see Figure $2 a),, the imputation of Z is based on the following decomposition:
. (Z A,A,YY (X
A, (Z
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Accordingly, the structure of the CI-StoNet can be arranged as follows:
Z μii (X, 01) ez:
A = n2 Z 02) + Ea:
(A1)
Y Ma X Z, A, 0z) + Ey:
where Es Eo and By denote Gaussian random errors. The corresponding diagram shown in Figure $2 b).
(b)
00O0 O
Y
Outcome
X
A
Treatment A
Z
Proxy
Figure S2: (a) Causal structure: outcome depends on the proxy; and (b) CI-StoNet structure for the scenario where outcome depends the proxy.
A2.2 TREATMENT DEPENDING PROXY
When the treatment depends on the proxy, see Figure S3 a), the imputation of Z is based on the decomposition:
A, X) T( Z) X (Y| (Z
The structure of the CI-StoNet can be arranged follows:
Z μ1 (X, 01) t
A = P2 Z, X, 02) + Car
(A2)
Y Nu N Z, A, 0z) Cy²
where e,, Co, and Ey denote Gaussian random errors. The corresponding diagram shown in Figure $3 b).
A
x
Y
Prexy
312 e.)
P.Y A.8.4)
Outcome
Treatmant
Z
Procy
Figure S3: (a) Causal structure and (b) CI-StoNet structure for the scenario where treetment depends the proxy.
Both models can be trained using an adaptive stochastic gradient MCMC algorithm, and the corresponding causal effects can be estimated based on the imputed confounders from
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For causal structures shown in Figures S2 a) and S3 (. X is the proxy variable, Z represents the missing confounder, and A and Y represents the treatment variable and outcome variable, respectively. The white nodes represent observed variables, while the light- grey node represent the unobserved variables. For the CI-StoNet structures shown in Figures (2(b) and [3),,, white rectangles represent variables from observed data; light-grey rounded- rectangles represent hidden neurons; and dark-grey rectangles represent network modules to learn respective conditional distributions.
A3 EXPERIMENT ON DAG MISSPECIFICATION
We conducted an experiemnt where the true data generating process treatment- -repndnnt-depedddng proxy and outcome- depending proxy, but we used basic proxy to fit the data. Following is data-gencrating procedure
1. Confounder: Z = (Z1, 22, 23, 21, 25) ~ 15)
2. Proxy: for E {l..... ,0), X, h2 + ex, where hz(Z) sin(21) + 0.5 +
0.3(23 + 21 + 25), cx ~ (0, 0.5l50)
3. Treatment: let hz) 0.721 0.322 0.2z3, g(X) = dx E Xx 1- For outcome
depending proxy and basic proxy, Bernoullíexpiit(Hy ()) For treatment-
depending proxy, A ~ Bernoulli(expit b2 (Z) + 0.5g(X))).
4. Outcome: let f(Z) 2 0.52223, r(Z) 3 + 0.5 (2)), (X) + Xاl- For treatment-depent dependen proxy and basic proxy, outcome-dependent proxy, Y = f(Z) + A ((z) + xxx) cy.
f()) + A +-)()))+yy For
N(0.0.53)
The true ATE is 3. For each scenario, 10 simulation datasets generated, each dataset contains 2000 training samples, 500 validation samples, and 500 test samples. We use the basic proxy structure (equation 15) to model all three scenarios, where basic proxy scenario provides a correct bascline while outcome-depending and treatment-depending provide demonstration of DAG misspecification. In-sample MAE is calculated across train and validation set, and out-of-sample MAE is calculated across test set.
Table S3: Mean Absolute Error of ATE estimation under DAG misspocification
In-Sample MAE
Out-of-Sample MAE
Basic Proxy
0.0682 (0.0193)
0.0702 (0.0207)
Outcome-depending proxy
0.0705 (0.0188)
0.0976 (0.0236)
Treatment-deeendiin proxy
0.0911 (0.0156)
0.1239 (0.0178)
A4 DIAGNOSTICS AND UNCERTAINTY QUANTIFICATION
Diagnostics on latent overlap Although Z is latent, we can still perform robustness diagnostics by sampling from its posterior. We show the procedure of diagnostics latent overlap under simple confounding setting. Given that we have a overlap threshold 0 < a < 1, and propensity score model é(A[Z), A simple stress test is:
1. For each unit i, draw Z Po ( YJ), b=1.
2. For each draw, compute propensities and a summary statistic of overlap
((b)
Ie <ao>>1aa
3. Report the mean S. S. B Σ. SO as diagnostic.
This diagnostic does not "verify" the latent overlap assumptions, which is theoretically untestable, but it allows practitioners to detect violations or fragility of overlap under posterior draws of Z, analogous to how overlap is assessed with fully observed covariates.
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Bootstrapped Confidence Interval As explained in the imputation-reguulaiizd opti- mization paper (Liang et al.) 2018a), the uncertainty in the "Monte Carlo average over imputed z" reflects only the uncertainty due to the missing data (i.e., Z), and does not coincide with the uncertainty of the causal effect itself. Estimating the uncertainty the causal effect requires additional methodology, such bootstrapping method to compute the observed information matrix.
To derive bootstrap confidence interval, consider the following computatiinnally-lighu bootstrap procedure:
1. Fit CI-StoNet on the full dataset and save the final pruned parameters. The parameters are saved as the warm start for all bootstrap replicates.
2. Draw a bootstrap sample of size n, i.e., sampling indices (f.,..,, 2 i.id Uniform(1,...
with replacement, then construct the bootstrapped dataset D [AAg. Xgy Yg)]f=r
3. Warm-start model initialization. Initialize model parameters for this replicate by copying 0.
4. Run short SGD + SGHMC imputation phase to let the parameters to adjust to D³.
5. Compute bootstrap ATE estimate. Use the trained parameter 0" calculate 7 - E
For the basic proxy scenario in DAG misspecification experiment (see details in A3 100 iterations of such bootstrapping are conducted for each of the generated dataset. Note that the true value of ATE is 3.
Table $4: Bootstrapped bounds for basic proxy setting
DGP seed
Lbnotstrap
Uhootstrap
0
3. 131365
3.0897
3.1581
1
3.0057
2.9758
3.0574
2
3.0074
2.9836
3.0720
3
3.1582
3.1066
3.2113
4
2.9718
2.9609
3.0404
5
3.0123
2.9774
3.0801
6
3.0515
3.0263
3.1113
7
3.0556
3.0128
3.0904
8
3.1846
3.1181
3.2119
9
3.0241
2.9787
3.0471
A5 THEORETICAL PROOFS
99 can be established through several steps. First, we show that the estimator (k)) (0(+) (x) obtained from Algorithm converges in probability to solution of ( denoted by 0" (see Lemma A1 discussion on how to address the non-uniqueness of 0 is followed. Next, we show that 0 is consistent estimator of 0", the true parameter vector of the sparse StoNet defined
The consistency of the estimator
(see Theorem A1). Building on
this result, establish the consistency of the estimator (see Theorem
D These results
are presented the following.
A5.1 CONVERGENCE OF (k)
To train the CI-StoNet using the IRO algorithm, requires that the full dataset is used at each iteration, making the algorithm difficult to scale up to large-scale neural networks. In
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contrast, the adaptive SGHMC algorithm can use mini-batch data in parameter updating. As shown in Liang et al. (2022) the adaptive SGHMC algorithm solves equation under the following conditions.
Notations: We let D denote a dataset of n observations, and let D; denote the i-th observation of D. For StoNet, Dj has included both the imput and output variables of the observation. For the CI-StoNet, D, includes the treatment and outcome, i.c., D; [A, Y.). For simplicity of notation, we re-denote the latent variable corresponding to D; by Zi, and denote by [D, (2,, 0) = - log π Di, 0) the negative log-density func- tion of Zz. Let Z = (Z1, Zz..... Zn). let
(21, 22,.... 2n) be a realization of Z, let FD(Z,0) 0) = Ei= fD. (Z. and let H 9) = Ve log ( A, O). To study the conver- gence of the adaptive SGHMC algorithm presented Algorithm T we make the following assumptions:
Assumption A4. (i) (Boundedness) The function FD(...) lakes nomnegative real val-
ues, and there erist constants A, B 2 O, such that (F, 0²) s A, B, IVeFD (0, 00)l S B, and [| 0'1l S B.
(ii) (Smoothness) FD(:.)) is M-smooth and H(..)) is M-Lipschitz: there ewists some constant M > 0 such that for any Z, Z' E R. and any 0, 0' E O
VZF 0) VzFb(Z' 0') S M Z'Il + M 0 0'1 |||
(VFFDZZ.0) VoFD(Z', 0')ll - M Z'|l + M| 0' ||,
(H( H(Z', 0')ll S M z'll + M| 0'|
(iil) (Dissipativityy) For any 0 E θ the function F (0) is (m. (ltbb-ddisss -dissipatiie: there erist some constants m 1 and b 2 such that {Z, V F Z, 0²)) 2 ml 2 b.
(in) (Gradient noise) There exists constant E 10. such that for any Z and 0, E V Z (D (2 0)l s (M2 + M 1 B²)
Assumption A5. The step size {7h JEN is a posituue decreasing sequence such that " + O and Σ km1 7h- = oc. In addition, let h(0) = [ (Z, 0)), then there exists ô > 0 such that for any 0 E O, (0 - h(())) 2 --000011 1-, and lim inf 26 7
+ " -1 > 0.
T641
%
Assumption A6. (Solution of Poisson equation) For any 0 E θ, = E, and a function
v(=)=1+ = 1 |||||, there erists a function μe on 3 that solves the Poisson equation po(z) (z))(=)= = =0) h(0), where To denotes a probability transition kernel with To)lo(z) = (0 z')'z', such that
HO =k+1) h(Ok)) + no, (#k+1) -To, No, (*x+1), k= 1,2,....
(A3)
Moreover, for all 0,0' E θ and E 3, we have [1 uo (2)ll s (2) and [|*)(2)]| I (=) for some constants S1 > O and S2 > O. Lemma A1. (Theorem S1, Liang et al. (2022)) Suppose Assumptions A4 Ao hold. For Algorithm 1 if we set = =Ce/(e, + and nh Cc/(c,tka ) for some constants a E (0, 1), C.>0, C. > 0, ce Z 0 and eg 2 0, then there erists an iteration ko and a constant Ao such that for any k > Ko.
(( 0 12) 1 Ao
(A4)
where 0% denotes a solution to Eq. 7 i.e., 0, E = (0 Vo T (0 = 0}: and
An = No + /6 ((MM2 + c2) + 2M + 3B2 + 1/2 for some positive constants A/- Co, and Cz.
Proof. Lemma A1 is restatement of Theorem S1 of Liang et al. 2022 and its proof is thus omitted.
Refer to Lemma S1 of Liang et al. (2022) for the derivation of the constants Co and Oz which indicate the dependence of the convergence of ((k) on the structure of the StoNet
4 As a consequence of the lz-convergence (A4), we immediately have [1)(t) - 2 0 as
k 20, where 2, denotes convergence in probability.
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Remark A1. For neural networks, it is known that their loss function is invariont under certain transformations of the connection weights, such as reordering hidden neurons within layer or jointly changing the signs scales of specific weights and biases, refer to, e.g., Liong et al. (2018b) Snd et al. (2022) for detailed discussions. As a result the solution On is not unique, and all such solutions unique solutions, defined by loss-invariant transformations. This equivalence class forms reduced representation of the parameter space, where each membber corresponds to distinct network (Le., not transformable inlo another via loss-invuriant operalions) and may have a different loss value. The consistency results established in this paper apply specifically to this reduced space of neural networks.
be viewed belonging to an equivolence class of
A5.2 CONSISTENCY OF On
A5.2.1
CONSISTENCY OF THE IRO ALCORITHM
The IRO Algorithm The TRO algorithm [Liang et al. 2018a) starts with an initial weight
(0) 0 and then iterates between the imputation of latent confounders setting 0% 00 (0
and regularized optimization for parameter updating:
Imputation: simulate zi (++1) from the predictive distribution:
(2, Vss ai cen) x π(2; ags 0 (o,) 2, 2,, . ,gg)
(0))
where & indexes iterations, and oē, (o ot)
Regularized optimization: Given the
data {V,, 2((+1) a.)
1, 1,2....,n], n), update 0 (1+1) by maximizing the penalized log-likelihood function as follows:
0 (1+1) arg max
log (B 2, (1+1) aa 0, 0c) log Pa., (0)}-
(A5)
The penalty function log Pa,, (0) satisfies some conditions (see Assumption A9 such that ① (1+1) forms consistent estimator, uniformly over iterations, for the working parameter
((++)) arg max Egi log 2a 0,0
(A6)
arg max / log z/a, 0, o (T( y. a. "0
"0. .o))< a. 0" o
where 0* denotes the true parameter value of the CI-StoNet model.
Consistency of Parameter Estimation The main proof for the consistency of parameter estimation is built on the theoretical framework developed in Liang et al. (2018a). Let (A, Y. Z) be the complete data, which is collection of observed variable and latent variables. Define
G, (001000) -
y, y, log 2/a, 0, 0,
t , a, 0" .0000",,))22 oz)yzdy
G a. (0) = log ( 2. a. 0, 0,.r),
= a, "0 o2)
Lemma A2. (Theorem 1: Liang et al. (2018ah) Let denote the total namber of iterations
of the IRO algorillo. Under mild regularity conditions (See Assumptions 1-3 in Liang el al. (2018a)), the following uniform law of large numbers holds for any T, with log (') = o(n):
supg supece Gn(( ā. (0) G..( 0 000
E 0.
(A7)
as the sample size
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Assumption A7. For each = 1,2,..., Gn e²)) has unique marimum (up to loss-ineariant transformations) at " for any 0, supoeor Gn )²)) exists, where
(e(e) = (0 E θ 0 1 < e}. Let ò, = -G, 0²) - supeeer Gn 0%)),
mi.. o2 holds.
Assumption A7 restricts the shape G, (010 around the global maximizer, ensuring that is neither discontinuous nor too flat. Given the nonidentifiability of neural network models, Assumption A7 implicitly assumes that cach is unique up to loss-invariant transformations, such as reordering the hidden neurons within the same layer simultaneously altering the signs or scales of certain weights and biases, see e-g., Liang et al. (2018b) and [Sun el al. (2022) for further discussions. Alternatively, the optimal solutions can be considered as belonging to an equivalence class, subject to appropriate loss-invariant transformations, with the uniqueness assumption applying to this equivalence class.
Furthermore, consider the mapping M (0) defined by
M(0) arg max Eg log (Y, Z|a, 0' 0'
As argued in Liang et al. (201 Sa) and Nielsen 2000), is reasonable to assume that the mapping is contraction, recursive application of the mapping, i.e, setting
0 ((+1) 0 M()
leads to monotone increase of the larget expectations Eyc log Dog((Y, 2,,, 0,) for t= 1, 2.....
Assumption A8. The mapping (0) is differentiable. Let (n) be the largest singular value of (M There crists a number p" < such that Pm. (0) S p" for all Q for sufficiently large and almost every observed sequence of A.Y).
Assumption A9. The penally function + log Px, (0) converges uniformly over the set (0() :t 1, ... as n + oo, where An is a regularization parameter and its value can depend on the sample size n.
Lemma A3. (Theorem 4: Liong et al. 20188a)) Suppose the conditions of Lemma A2
Assumplions
A7 A9 hold, and sup,,t 0 < do hold. Then for sufficiently large and
almost every (A, Y sequence, 110% 0' 2 0, as
A5.2.2 VERIFICATION OF ASSUMPTION
A9
To verify Assumption A9 we prove the following lemma.
Lemma A4. Let 0 =
(0,00 (0,,02..... 0x, "x,) )², Suppose that all components of are priori
independent and they are subject to the following mixture Gaussian prior O Suppose K, > n. O is sparse at a level of mn X clog(K../n) for some constant c > 1, and min[|0
0, # 0,i 1,22 1. ..... Kn} 8n for some constant d = o(1). we set σı O(1) and set (o) to satisfy the conditions:
n
n
) - Am
<
Kn
K
(A8)
8,
)
- oo
min
1 {
Kn
K Velog(Kn) - -(c 1) log
then the following result holds:
1
log x (0) + Kn, log (V2Tou)
+0, as n + oc.
(A9)
n
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Proof. A straightforward calculation shows that
J0dn
Jodn An
(0) + Kn log(on)] S Kn. log(1 - An)l + (Kn -ma)
+ mn
log
01 2m)
mn 2n)o1
(
- mn 2
+
An
To ensure Kn, log((1 - AA] < n, we set
An x 1 k en/Kn n
(A10)
Kn
To ensure Mn log
ooda 1-dn
x n2, we set
00 >
)c > E En/men
An x
n
je= x e- -n/ma
(A11)
Kn
Kn
To ensure (Kn - -mx) (1(1-
AoAn
x <n, we set
Xn)
0u 1
(A12)
Kn
Kn An
62 7
To ensure m. (1(1-1.)0
Ando
0
"1
X n2, we set
on
8n
00
(A13)
Velog(Kn) (c 1) log(n)
log(nAn/mm,)
8" x n.
Since 8n x (1) and my x we have my 20
As a summary of A10 A13 we can set (An: Jo) as stated in A8 which ensures (A9 holds.
Theorem A1. Suppose the regularity conditions give in Lemma A2 and Assumptions A7- A8 (given in Supplement A55) hold. Additionally, assume that the dimension of 0, denoted by Kn, increases with n in a polynomial rate K, = O(n5) for some constant <> 1, while the true StoNet is sparse with the number of nonzero connections m, < clog(Ka//] for some constant c >> 1. Set the hopper-parametery of the prior
O to satisfy the conditions:
n Kn
An x
(2)"" + 0000 min ( K..' σn 0(1),
n
n K., Velog(Kn) (c log(n)
on
(A14)
Then 10,, 0' R. 0 holds as - 00, where 0* denotes the true parameter of the StoNet and 0 is up to a loss-invariant transformation.
Proof. Since 0% is a solution to equation it serves the maximun posteriori (MAP) estimator of 0 with respect to the incomplete data (by treating Z as missing). By Lemma A3 we immediately have its consistency with respect to 0*, i.e.,
0 0-
3 0, as 1 X0.
(A15)
Among the conditions of Lemma are gencrally satisfied. Recall that we adopt the mixture Gaussian prior (6) in computing the MAP of O. By Lemma
A3
we only need to verify Assumption A9 since the others
A4 Assumption
Ag is satisfied. This concludes the proof.
Remark A2. In Theorem A1 we assume that the true sparse StoNet is of size m,, = ((n)
This assumption can be justified based on the theory established in Böleskei et al. {2019) Schmadt- (201)) and Petersen & Voogglaender (2018), where it is shown that DNN of this size has been large enough to approrimate many classes of functions, including affine, piecewise smooth, and a-Hööler smooth functions. See Sun et al. (202g) for discussions on this issue. Additionally, [Snn et al. (2022) showed that a sparse neural network of this size has been large enough to achieve the desired function approzimation and posterior consistency, with the mixture Gaussian prior (6) as the sample size n becomes large. Our theory allows K,, to increase polynomially with n, which is typically satisfied by deep neural networks.
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A5.3 PROOF OF THEOREM
To justify the pooled Monte Carlo average, fix an integrable test function & and define 6i às Σi sz2("). Conditional on A; di, the draws 2; (1) p(- a;) are ii.dd (or ergodic), hence
Justification the estimator (9)
P 4 (=(())) a:)
as M xc,
where 5 denotes convergence in probability. Therefore, for any fixed n,
M
(z
P1
Σ B(p(z) a.): as M - oc.
n.
Under Assumption {a; ) can be assumed to be i..d., the weak law of large numbers implics
7 [[p(z) a.} 4 Ex(E(p()) a}l = / ((z)
as + 0C.
Proof of Theorem
Proof. Consider the joint density function:
TZ, 0") = x(Z)A, AY, 0,),),
under the assumption that the true model is a sparse StoNet + parameterized by 0", Then
have
B(Y(a))0)) = (z(2) 05)dzdy = pplzz a, (z)
(1) Let 2; for -= 1,2, ,M,,, denote M independent samples drawn from क (( a, 01). Let
((Y nM (2 a. 07).
By the standard property of Monte Carlo averages, as justified for the estimator 99 we have
[(Y(a)))))) ((Y(a)) 1 0, as n. M + co.
(A16)
On the other hand, by the consistency of 0% 07) (with respect 0*) as established in Lemma A1 we have
[E((())) B(Y ||| 3 0, oc,
(A17)
since 22(.)) is continuous respect to the parameters (as assumed for the neural network model).
Combining the convergence results in A16 and A17 we have
[[E(Y
< ([)((())) E(Y 0
( (1)1)EE())))) &
as + co and M + so. This concludes the proof.
A5.4 PROOF OF THEOREM 2
We can leverage the theory developed in (Sun et al. 2022 justify the misspecification error of the causal effect.
Assumption A10. (Restatement of Assumption A.2 of Sun et al. 2022
The sparse DNN
model AAd. and n. satisfy the following conditions:
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1. The network structure satisfies:
Tn logn + Tn log L + Sn log Pn I Cont=s,
where 0 < E is small constant, In hr denotes the connectivity of r" == maxIsisHH-114 Ljj denotes the marimum hidden layer width, and 3n denotes the imput dimension of n
2. The network weights are polynomially bounded:
10 S Ens
En n1
for some constant C > O.
piecewiso-smooth functions with and bounded a-Holder smooth function (Polson & Ročková (2018)). Assumption A clarifies the class of sparse DNNs that can approximate the unknown structural mean lunctions, and hence considered as "true sparse DNN" in the paper. Informally, Assumption A10 requires that the sparse deep network whose number of active weights and relevant inputs grows slower than the sample size, and whose weights are at most polynomially large in n. This ensures the "true" network lies capacity-contrroled function class. By changing the network structure and network weight accordingly, we can establish different approximation error upper-bounding sequence wn for different function classes F, see discussion Sun et al. (0022) Section 2.2 for more details.
For example, affine-system functions Bolcskei ct al. (019
fixed input dimension (Petersen & Voigtlacnder (2018
Proof. Let In denote the class of sparse DNNs compatible with the Assumption A10 we define for each sample size n the pseudo-true sparse DNNs (Y, as minimizers of the approximation error within Gn:
(K E arg min
(llMA,- - mAll 12 + |ay, my 11
(MI.P))EG.
by Assumption 3 there exists sequence wn - 0 such that
{llAA - -allza + || my S Wn
with Wn scaling at the chosen rates for chosen function F (e.g. Wn x n a/(aatd up to logarithmic factors for Hölder functions in dimension d).
Let Po denote the true joint law (A, Z,Y). For any parameter y of sparse DNN, let Q, be the induced joint law under the corresponding generalizod linear model (Gaussian logistic) with regression function (,,, and let Po be the distribution induced by the CI-StoNet with parameter O. Following Sun et. al. (2022), we define Kullback Leibler divergence
/ goog
q p
do(g.
and a family of distance de(g,p) = f ((())) - 1] for any t 0, which decreases to do as decreases toward O. For Gaussian regression with mean functions my and m2, they show that, for fixed noise scale and bounded regression functions with known o²
da(pm, Pmm)
20 (m² m2)²
and for logistic they derive upper bounds
1 [ - =m + 0(m0 - -m2)3)
di (pm, Pma) S
Since do(Pmn Pung) di (Pmi Pmg), we have do(pm, Pmm)) I C(m 2 for constant C' > O depending only on the uniform bound on μ.. Then it gives
[L.(Po, Qn) S 11,, 1112
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where My is the DNN-based regression function in Qn and μ²" is the true regression function. Putting things together, if the function class F is approximable at rate wn in L2 by sparse DNNs, then there exist sparse DNN parameters 7,, such that
KL(Po: Qn, S w
In the StoNet definition, the deterministic parts mı, ma themselves neural networks. Therefore, for any sparse DNN architecture that satisfies Assumption [A10] we can embed the same architecture into the StoNet by choosing that the StoNet's deterministic part coincides with the DNN, which yields a mapping - O(T) with On = Pe(n) Therefore, defining pscudo-true StoNet parameter
0 arg min (L,,
then we have
(L.Po. I KL(Po. (L(( = KL(P), Q,) S w
Now consider the simple confounding case (recall the setup in Let (PP) I my (a, 2)pr, (z)dz and (Poo) = (2(2, x)pp. (z)dz, the approximation error between the true and the pseudo-true causal estimand is:
- (Po )1 =1| f my 2))r,)))))( -
f pz(a,, 2)pro (z)dzl
S (my((,,) p2(a, 2) lpro (z)dzll + f v2(a, 2)(pro pr (z))d2)
A2
A1 can be bounded by
A S || (a, 2) - 12(a, (lla(()))))) S wn.
The first line is by Cauchy Schwarz, and the second line is by sparse-DNN approximation results and StoNet/DNN equivalence.
Assune (22(a, 2) S Cpar then
A2 5 Opa / (Pro(e) - po- (2) dz
1 1 |KT.(Po.2. .o.
S 20
S V2C, KI,
S wm.
From first line to second line we use Pinsker's inequality, and from the third line the last line we used the previous result that KL(Po, Po-) S w3 From the second to the third line use the data- mata-processing inequality for KL divergence (monotonicity under marginalization):
(L Po 2) < KL,
Putting pieces together, we have
[pb(P)) - (P & A1 A2 Wn + Wh S Wm.
Hence we propagated the approximation error of muisance functions to the causal estimand and proved that the approximation error for causal estimand is O(w,).
Theorem in our paper controls the estimation error, which is guaranteed by the properties of the sparse DNN. The misspecification error is determined solely by the approximation power of the sparse- class (depth, width, sparsity rate). So the pirors only impact the misspecificaiton error indirectly, through controlling the model capacity.
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A5.5 FINITE-SAMPLE ANALYSIS ON OVERLAP INFLATION
Assumption A11 (Boundedness and overlap). For each treatment a E A, let m (2,a)
denole the true oulcome vaiiaane and mlzza)) a) ils eslimalor. Assume:
1. Oulcome boundedness:
sup Im (z, a)l Cms
sup (m(2 a)) Cm.
2. Latent propensity (positivity) bounds: there erist constants 0 < (iz(a) S (a(a) < 00 such that for all in the relevant support,
K₂ (a) S v (a))),
<2(a) S f(a) 2) Kz(a).
3. Observed propensily bounds: there exist constants 0 < kzz(a) S Caz(a) < oo such that for all prozy Dic
(a(a) S p' x4) Caz(a), x²(a) S p(a x;) S Caz(a).
4. Marginal trealment density bound: For each treatment value a E A, there exists KA(a) 0 such that
p'(a) kA(a),
p(a) 2 kA(a).
Define x<(a) := min((n mmn((xx(0)) (a)( k,(a), (a)) and K(a) == (a).
Lemma A5 (Bayes-bridge in simple confounding). Fir A with p(a) > 0. Then
p(a)
p(z) = p(== [
aa a),
w(2,a) = p(a 2)
In particular, under Assumplion A17
K(a)
(w)2za)= 11 max {1
KA(a) K(a)
0 w(2, S The same stalements hold for the estimaled model with (( a) //aa)/((a 2).
x((a)'
K(a) x(a)
Proof. Bayes' rule gives p(z a)) p(alz)p(z)
p(a) hence p(2) = p(2 ()) p(a)
pla|=)- Under Assumption
A1l p(a =) > x(a) and p(a =) < K(a). Also p(a) f p(a =))(z)dzz < K(a), and by Assumption A11 (4), p(a) 2 (a). Therefore w(z.a) - K(a)/k(() and w(z, a)
<A(a)/K(a) which implies the bound | w/-1/.
Fir a E A. Let m (z, a) E|Y A a. Z and f a) p2 (z, a: 0z). Suppose Assumption A11 holds. Define the population causal estimand
Theorem A2 (Simple-confounding bound for estimator
(a) = / m (z, m2 a) (z)
Assume the latent draws used in satisfy
i.s.d. p(z as): i= 1.....,, 1 1= M,.,,,
conditional (a.) and the fitted model. Consider the estimator
M
(a)(a)= nM 1=1 1=1
m a).
Let the corresponding (model-based)) conditional plug-in targel be Dn,cond (0:a): = n i=1
Σ
/
m(z, a) $(2 a;) dz,
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and rewrite v² (a)
3² (a) 4 bridee (a)
m" (z, a) p² a,) w az, dz.
i=1
where
w (z ai) = pa(aa:
p" (at 2)
For any ô E (0, 1), with probobility at least 1 - 8,
1og(2/5) (() - (a)) s Cm
s nnnan (0:a) x noud (a)
nM
Monte Carlo
estimation ander p(z|a)
T
nn (a) t bridge (a)
Bages-bridge (overiap-inflated) bias
where Cm is from Assumption A11 (1), and
1 Σ
cn (a) = ni
m² m= a.) dz.
Moreover, the Boyes-bridge bias term satisfies
Σ
/ m² (=, ai) (2, - 1) dz
$ (.cood(a) $6 "..brddge (a)
m
{1- kA(a;)
K(a;)
I
C
max max SSPS
K(a:) <<(a))
Proof.
We prove the claimed decomposition and bounds.
Step 1 (error decomposition). Add and subtract Pn,cond((; a) and o (
(n) 3i (a)l (1n - x; ba)d
(a)
( 4 nn,cond a)l n (0: a) " " (a)l + Φ "n.cond (a)
+ bridge (a)).
This gives the displayed three-term bound once each term is controlled.
Step (Monte Carlo term). Conditional on (a:) and the fitted model, by assumption the draws 2(!) i.dd. p(z aa)) are independent across (i, 1), and mn(=()) (1) a) [-Oms Cm.). Thus, Hoeffding's s incquality implics for any > 0,
P P((in.ula) T Pn 0 a) > >c) exxp
nMe3
20
Setting the right-hand side to ā yields that with probability at least. 1 5,
2 log(22/)
(a) T (0 a) s Cm
nM
Step 3 (Bayes-bridge bias bound). By definition,
T " (a) - -ii bridge (a) = Σ
/
m m² (z) (s) (1- dz.
1=1
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Taking absolute values and using (lm="(, a) I Cm.
nn (a) tn bridge (a)
Cm <
n Σ
/ p0 ai) 10 (2, & Su-supp |w (2, a:)-
Under Assumption All for any a and all x(a) < (a =) - K(a). Moreover, pa(a) = S (z)dz [(),, and by Assumption A11 (4), pa(a) 2 (a Hence,
KA(a) K(a) pa x) k(a)'
I pa(a)
K(a)
so for any i (with a, place of a),
p" (a.) pa 2)
S max { 1- {1- K(ai) k(a;)
ka(ai) K(a;)
}
sup I a:) 1| sup
-1
Step 4 (estimation-uudder-pllll term). The remaining term Pn,cond (0; a) $ e. (a) is exactly the estimation error of the plug-in functional under the conditional
latent laws:
t nn (a) 1 Σ
n. cond (0: a)
[miz a)D(= a.) - -m) mz (z, a)v" (z a.)]
If desired, it can be bounded further by adding and subtracting m m= a)p a.) and using ||m* (s +))l S Cm, Iff(..)))la & C.:
onnd (0;a) (n,cond(a)) S (a) + 20m n
[ī(z a:) p² a.)
where 6m (m(a) sup, jm(z a) m² (zx a)(.
Combining Steps 1-3 yields the stated three-term high- high-probability bound, and Step 3 provides the explicit overlap-inflated Bayes-bridge bias bound. This completes the proof.
Assumption A12 (Proxy conditional independence). In the basic prozy selling, assume
A 1 X Z:
equivalently p(a 2, = p(a =) for all (a, z, x).
In consequence, we have the proxy Bayesian identity:
p(a x) p(z xz)
pl ax
(A18)
p(a x>)
Lemma A6 (TV bound for proxy conditional p(z x. a)). Assume Assumptions A11 and
A12 Fix æ. Let f(a, z|x) : p(a z)p(z x)) and f(aa z|x) = i(a =))(z x), with
normalizers = p(a x) = f fa x x|x) dz and 9 = p(a x) = f(a, 2|x) dz. Then
|lf [[β(2| x. (z x. a)ll S 2 (a x)) I1f-filn f1h+ (a x) p(a x) |M x)) (a x)).
Moreover, using Assumption one has the crude bound
|lf -f- S ((a) [ x) px x))11 + Ep-(m)) -)- pp x)]]
Proof. Write p( x, a) fa, x x/x)/0 and p' a) f(a, x/x0// with و pr (a x)
and 9 = f(a x). Then
f f
f
6 / 1), = 1|f - fll + 11
1
<
+
0
9
g
9
1
1
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Using 1/8 }
او-ها and dividing by 2 yields the stated TV bound. For the crude bound 9B
on Ilf ffl., add and subtract p(a xx)p" (z w):
Ilf - flla f p(a x) 10( æ) p" (= x) dz + / p'(x ab(a x) - -p"(a x)) dz,
and apply p(a 2) < K(a).
Assumption A13 (Gaussian latent working model stability). Suppose the CI-StoNet working model for the latent Z is Gaussian with fixed covariance:
p(z u; 01) NNm (w; o
where for prozy, L.e. p(= 01). Assume M1 is uniformly Lipschitz in 01:
| (w; 02) - (u 01 S 1110; - 0ill.
Vu.
Lemma A7 (Pinsker Gaussian TV bound). Under Assumplion
A13 for any u,
L
||p[ u; 01) - -p(-1 0illly 2
110
0ill
20,
Proof. By Pinsker's inequality, -P-QQITVV S (P1(())) For Gaussians with common
covariance I,
11 - -
(1
2
Hence
1 ||~e 2
|1
All2
Li
110
-i1l:
20
20
20
where the last step uses Assumption
A13
Theorem A3 (Finite-sample bound with explicit overlap inflation). Fix a E A and suppose
Assumptions 111 A13 hold. Let the sample-conditional causal target be
(a) 1 Σ (aY ai] = Σ
/ m² (z, x' xx;) dz.
Define the plug-in funclional
4 2r (0:0aa = 1
n Σ
m(z a) p(z xx)) dz,
and the Monte Carlo approcimation
M mi , .).
(a) = nM
z,
s.i.d. fi xz))
i=1
Then for any d E (0, 1), with probability at least 1- 8,
2 2log(2/0) nM
(a) xa)a S C.
+ (0;((;a) (a))
Moreover, define
w/ (z, a) = (a xz))
(x(2.a) = $(a x;)
(a 2)
p(a z)
Can (a) = 1
(a) = sup |ā(a z) - pa x)),
li(a xx)) - (a xx))
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Then
20m Car 1 n
(0: (a)( x
E
|í(z a; (z De a)llTv
k(a)
distrhuatoon/reonnttutiiio error (overlap inflated)
Caz k(a)
Cm
Ear (a) + Caraa
Cm (a)
k(a)² Eas (a)
<(
outcome nuisance (overlap inflated)
weight error foverlap inflated)
Proof. We analyze the error ( - (a),,,,
(a) (a)) I ( (a) t a) A
(0:(0:a) a) & (a)
where A1 represent the the error attributed to Monte-Carlo approximation, and Ag represents the error attributed to estimation of nuisance function. By Hoeffding's inequality, have,
nMe?
P(A1 o) 5
2C2
Equivalently, with probability at least (1 dd),
2log(2/0)
A S Cm
n.M
By A18
have
p'(z a.)) = p=(22 1 m; a) p²(a xx))
p" 2z ()
pa 2)
and similarly $(z x.) = f(( x a) (z, a). Therefore
(a) n
m" a²(zxa)p'(2 x 24, wf (z, a) dz,
2 ( a)
ml a) a(z æi ú, a) dz.
For the term Ag, we have
A₂ < Σ
/ (m - m² piaut) dz)
t=1
where Pio >p(z Di, a) and Pia p²(2 I a). Add and subtract in ph w. to get
Ag < B1 + B2 with
Bi Σ / (0)2, pia (=)) dz
Σ / (1 - (z) a)) (z) dz
B2 = n
Bound Bi. By Assumption All
(1) < p(a xx) Caz(a)
inf, piaa) z) x(aa
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Thus
|m (l) - -iill 20m Car(a) k(a)
B1 <
Pial/rv.
1=1
1=1
Bound B2. Decompose viū;; - mm" " = (m - m=) + m (ni; w.)
For the first part,
[ m )w] pia dz
Caz(a) x(a)
| 1bo Epg [lm- m²
cm (a).
For the second part, using 1 -
-u-ul and Assumption A11 (nf
p(a xx)) (a xx)
[00 a) - -w; (z, a)l
f(a z) pa(a [))
iā(a æx) - pp (a ax))
Caz(a) k(a)²
pā(a | 2) -p"aaa x)).
x(a)
Thercfore,
J m (2, (z. (Bi(z.))) - (2, (..aa)pia( (z)
Cm x(a)
Eax (a) CmCau(a)
S
Eus Eas (a).
x(a)²
Combining the bounds for B1 and Bg gives the stated proxy inequality.
Remark A3 (Overlap inflation). In Theorem
A3 the samile bound contains
plicit inverse-overlay factors 1/m((a) and 1/k((a) (a(a)² so for fixed muisance estimation errors (Gas Caz, Caz) the bound worsens k((a) 4 O. In addition, Lemma A6 shows that controlling [ æ, (z æ, avllTv can introduce multiplicaative dependence on K K(a)= = sup. p(a =)
(and its estimated counterpart), which is particularly relevant when A is continuous.
A6
EXPERIMENTAL SETTINGS
A6.1
SIMULATED EXAMPLES
A6.1.1
MISSING CONFOUNDERS
For case with missing confounders, the hidden layers of the network consists of two modules. The first module lakes the treatment variables as input and imputes the latent confounder, and the second module takes the concalenated vector of the imputed confounder and the treatment as input to model the outcome. For separable confounding and non-separable scenario, the first module contains two layers with size 32 and 6, and the second layer contains two layers with size and 4. The variance of the noise term e, and ey in (4) are set as 10-5 and 10-3, respectively. The training consists of three stages pre-training, training, and finetuning after pruning, with epochs being 100, 500, and 100, respectively. The network is trained like a plain vanilla DNN for pre-training and training, but the decay of imputation learning rate e and network parameter learning rate % only starts at training. After training, the network is pruned and refined during the fine-tuning stage with smaller learning rate. Finetuning stage is usually optional and doesn't 'tt have dramatic improvement to the overall performance.
The initial imputation learning rate is set at 5 x 1 for non-separable confounding and e 10-3 for separable confounding, and decays with E; = Ttex The initial parameter learning rate 7 is set 5 10-7 and 10-6 for the first module and the second module, respectively, and decays with 7 1
76 For the mixture Gaussian prior A, = 10-6
of = 10-4, and of 10
30

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Published as a conference paper at ICLR 2026
A6.1.2 PROXY VARIABLE
For case with proxy variable, the hidden layers of the network consists of three modules. The first module takes the proxy variables as input and imputes the latent confounder, the second module takes the concatenated vector of the imputed confounder as input model the treatment variable, and the third module takes the treatment variable as input and model the outcome. The first module contains two layers with size 64 and 32, the second layer contains one layer with size 16, and the third layer contains one layer with size 8.
The variance of the noise term Cz, Cas and Cy in (2) are set as 10-5, 10-1, and 10-3 respectively. The training consists of three stages pre-training, training, and finctuning after pruning, with epochs being 50, 100, and 50, respectively.
The initial imputation learning rale are é1 10-3 and 62 = 10-4, and decays with e fa. 14c The initial parameter learning rales are set as 71 = 10 10-6, 72 =2=5 =5x 10-5, and
7' = 5 x 10-?, for three modules, respectively, and decays with Y 1+7xx00.0 For the
mixture Gaussian prior
66, An 10-6, oß 10-4, and of = 10-2
A6.2 BENCHMARK DATASET
The network structures for benchmark dalaset is similar to proxy variable.
A6.2.1 ACIC
The first module contains two layers with size 64 and 32, the second layer contains one layer with size 16, and the third layer contains layer with size 8.
The variance of the noise term G2, Cas and ey in 12 are set as 10-6, 10-4, and 10-3, respectively. The training consists of three stages pre-training, training, and finctuning after pruning, with epochs being 50, 100, and 50, respectively.
The initial imputation learning rate 61 c I+c The initial parameter learning rates are set as 71 = 11-66 72 = 10-5, and
10-3 and e2 = 5 10 and decays with
1s 10-7, for three modules, respectively, and decays with n 1+axx For the mixture
Gaussian prior (6) A, = 10-6, of x 10-4, and of = 10-2
A6.2.2 TWINS
The first module contains two layers with size and 32, the second layer contains one layer with size 16, and the third layer contains one layer with size 8.
The variance of the noise term es, eas and ey in 12 are set 10-3, 10-5, and 10 10-7, respectively. The training consists of three stages pre-training, training, and finetuning after pruning, with epochs being 100, 1000, and 200, respectively.
The initial imputation learning rate are 61 3 x 10-3 and = 55x 10-5, and decays with
6g = I+. The initial parameter learning rates are set as 71 = 10-3 722 = 10-5, and
7/ = 10-10, for three modules, respectively, and decays with 7k = 74
T4m x60.5. For the
mixture Gaussian prior
66, An 10 6, of = 2 10 and of 10"
31


Reviews:
[{"review_id": "pxDQG9VvQy", "reviewer": "Reviewer_NAnQ", "summary": "This paper proposes CI-StoNet, a latent-variable framework that treats missing confounders as stochastic hidden states and jointly models (i) their conditional distribution and (ii) the outcome with two interconnected neural networks. Training alternates between imputing the latent confounders via adaptive SGHMC and updating network parameters, enabling consistent causal-effect estimation even when the confounders themselves are non-identifiable up to loss-invariant transformations. The approach targets complex, nonlinear settings; retains identifiability of causal effects under mild assumptions; and extends naturally to multiple-cause and proxy-variable scenarios. The authors prove convergence of the adaptive SGHMC estimator and consistency of the resulting causal-effect estimates, then demonstrate state-of-the-art accuracy on simulations and benchmarks (ACIC 2019, Twins). They note limitations around DAG specification (e.g., mediators/colliders) and current lack of explicit uncertainty quantification, and outline how CI-StoNet’s Markovian structure can be adapted to broader proxy-based causal graphs.", "questions": "Please refer to the questions in Weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper offers a strong conceptual framing for missing confounders by treating them as latent states within a StoNet and jointly learning their distribution with treatment and outcome modules.\n- The proposed method reaches state-of-the-art ATE accuracy on ACIC-2019 and performs strongly on the Twins dataset; performance remains comparatively stable even when a key confounder is removed, indicating robustness of the latent-imputation mechanism.", "weaknesses": "1/ Your inference of $Z$ conditions on each datum’s $(A_i,Y_i)$. When both variables are binary while $Z$ is multi-dimensional, recovering $Z$ from $(A,Y)$ alone is under-determined; the latent confounders are generally non-identifiable without extra structure or additional observables. Could you formalize when $(A,Y)$ contain \\emph{enough} information about $Z$?\n\n2/ You state that causal effects remain identifiable “under mild conditions” even if $Z$ is only identifiable up to loss-invariant transformations. Please state these conditions precisely for the binary–binary case and add guidance about what fails if they are violated. Under what restrictions on $g_1,g_2$, noise, or overlap does $E[Y(a)]$ remain identifiable despite non-identifiable $Z$?\n\n3/ Because $D_i=\\{A_i,Y_i\\}$ drives the imputation of $Z_i$, the method’s success hinges on how informative $(A,Y)$ are about $Z$. Please articulate minimum “richness” assumptions (beyond overlap) under which the estimator is well-posed—e.g., conditions ensuring that $\\pi(Z\\mid A,Y)$ is sufficiently concentrated to support accurate effect estimation.\n\n4/ You provide proxy-variable extensions for “outcome-depends-on-proxy” and “treatment-depends-on-proxy.” It would help to add practical criteria or diagnostics for when $(A,Y)$ are too sparse and $X$ must be incorporated (e.g., signal tests or ablations showing failure without $X$). Could you include a short decision guideline indicating when to introduce proxies?\n\n5/ The convergence of adaptive SGHMC is asserted under specific conditions and in a reduced parameter space (due to loss-invariant transformations). In practice, what diagnostics should users monitor to assess mixing/convergence and to detect cases where $Z$ is poorly informed by $(A,Y)$? Provide a checklist (e.g., effective sample size, potential scale reduction, stability of $E[Y(a)]$ across chains).\n\n6/ You already conduct a robustness test by removing a key confounder in \\emph{Twins}. Please add a complementary experiment where $A$ and $Y$ are binary and $Z$ is high-dimensional, to quantify: (i) degradation in effect estimates, and (ii) the (non-)recoverability of $Z$. How does performance change as $\\dim(Z)$ increases under fixed binary $(A,Y)$?\n\n7/ Your ATE estimator is a Monte Carlo average over imputed $Z$, which suggests posterior uncertainty is available. Please report credible intervals (or bootstrap CIs) for $\\widehat{E}[Y(a)]$ and $\\widehat{\\tau}$, and compare coverage when $(A,Y)$ are low-information versus when proxies $X$ are added.\n\n8/ Given the potential information bottleneck when $(A,Y)$ are discrete, practical users would benefit from sample-size versus complexity guidance. Please provide rules of thumb linking $n$, $\\dim(Z)$, and network sparsity/width needed to stabilize effect estimation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7C0saRDxlV", "reviewer": "Reviewer_mx4M", "summary": "This manuscript proposed to estimate the causal effects with the missing confounder through a stochastic neural network. The authors provide some preliminary theoretical justifications, with some experimental justifications on both synthetic and small real world scenarios.", "questions": "My questions are mainly two-fold:\n* Think about the case that we only have $(A, Y)$, where $A$ is just some binary treatments, then I don't think we can perform reasonable ATE and hence if the conclusion claimed in this manuscript is correct. The only explanation is that we embed some inductive bias in the function class of $\\mu$ (similar to the consistency claim in Theorem 2 that depends on $\\theta^*$). If that's the case, we hide the missing confounder issue by adding additional identifiability issue on the function class. Is that correct?\n* Regarding the theoretical guarantee, if the density ratio $P(A|X)/P(A)$ is negligibly small but not zero, then we will have a extremely bad convergence rate. However, Theorem 2 only provides asymptotic convergence, and I'm generally wondering the dependency of some other quantities like the one I just mentioned. Otherwise it's not a real causal effect estimation solution but just an attempt to directly apply the stochastic network's results to causal effect estimation.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Overall the authors present the idea in a reasonable manner, with reasonable improvement on the performance.", "weaknesses": "The assumption of the existence of the underlying stochastic neural networks as inductive bias can be a chick and egg problem on identifiability. Meanwhile, the theoretical justifications are mainly on consistency, instead of convergence rate based on some standard assumptions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Nx10Rhsvto", "reviewer": "Reviewer_4cfB", "summary": "This paper proposes a new latent variable modeling approach called Confounder Imputation with Stochastic Neural Networks (CI-StoNet) to handle missing confounders when using observational data. CI-StoNet leverages stochastic neural networks to jointly model outcomes and unobserved confounders, and employs an adaptive Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) approach for both imputation and parameter learning. Theoretical guarantees are provided for convergence and consistency, and experiments on both simulated and benchmark datasets are conducted to confirm the effectiveness of the proposed method.", "questions": "Please refer to Weaknesses 1-3.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The paper tackles an important problem (i.e., missing confounders) in causal inference and proposes a new neural network-based method (CI-StoNet) that overcomes some limitations of prior latent variable approaches, such as limited applicability to nonlinear models and consistency issues.\n2. The authors provide theoretical support for the convergence and consistency of the proposed method.\n3. Experiments on both simulated and benchmark datasets are conducted to evaluate the performance of the proposed method.", "weaknesses": "1. As noted in the limitations, the method assumes that the underlying causal structure (DAG) is correctly specified, which may be difficult in real applications. Is it possible to provide some experimental results when the DAG is misspecified?\n2. The approach involves training deep neural networks with adaptive MCMC, which can be computationally intensive. Complexity analysis or comparison with baselines should be provided.\n3. The two types of baselines are inconsistently used in Table 1, S1 and S2. The differences and reasons should be clearly described.\n4. The analysis in the paper relies on several technical assumptions, which may not always hold.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "F8KGWRx4af", "reviewer": "Reviewer_mrwL", "summary": "This paper proposes CI-StoNet (Confounder Imputation with Stochastic Neural Networks), a deep learning framework for causal effect estimation when confounders are unobserved. The key idea is to model the causal DAG using a stochastic neural network where missing confounders Z are treated as latent variables and imputed from the conditional distribution π(Z|A,Y). The authors use adaptive stochastic gradient Hamiltonian Monte Carlo (SGHMC) to simultaneously impute missing confounders and train the neural networks. They provide theoretical guarantees showing that causal effects remain identifiable even though the missing confounders can only be identified up to loss-invariant transformations. The method is evaluated on simulated data, proxy variable settings, and benchmark datasets", "questions": "1. Why is stochasticity needed and why is it helpful to identify causal effect with missing confounders? Is the stochasticity in Z essential, or just a convenient modeling choice?\n2. How do you determine if the causal effect is identifiable? According to Theorem 2, causal effects are identifiable when assumptions 1-2 hold (including assuming true model is a sparse StoNet and conditions of Lemma 1 and Theorem 1 hold). However, these are not practically verifiable. A practitioner cannot check if the true model is a sparse StoNet. \n3. How do you determine if confounders (missing) are non-identifiable? The paper claims confounders are non-identifiable due to \"loss-invariant transformations\" but then states causal effects are identifiable (Theorem 2). Remark 1 mentions equivalence classes, but how does one determine which equivalence class the learned confounder belongs to? Can you provide examples where two different confounder values yield the same causal effect estimate?\n4. How realistic is the mixture Gaussian prior assumption? This assumption in Equation 6 is quite strong. Assuming all θ_i are independent is unrealistic for neural network weights, which typically have complex dependencies. Also, why is this two-component mixture appropriate? No justification provided\n5. The conditions in (13) are complex. How sensitive are results to the choices of your hyperparameters?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper provides convergence guarantees (Lemma 1) and consistency results (Theorems 1-2) for the proposed approach under sparse deep learning theory, which is a notable contribution.\n2. The Markovian structure of CI-StoNet allows modeling diverse causal structures, including multiple causes and various proxy variable settings (Sections 3, A2).\n3. The paper makes an interesting observation that causal effects can remain identifiable even when confounders are only identified up to loss-invariant transformations (Remark 3).\n4. The paper includes simulation studies with both separable and non-separable confounding, and evaluates on multiple benchmark datasets with comparisons to many baseline methods.", "weaknesses": "1. While the paper uses \"stochastic neural networks,\" the role and necessity of stochasticity is not clearly explained. The model (4) appears to be a standard neural network with additive Gaussian noise. Why is this stochastic formulation necessary versus standard deterministic neural networks with probabilistic inference? The connection between stochasticity and identifiability of causal effects needs clarification.\n2. Assumption 1(ii): Assuming the true model is exactly a sparse StoNet is very strong and unlikely in practice. This eliminates model misspecification concerns but is unrealistic.\n3. Mixture Gaussian prior (Equation 6): The choice of this specific prior with independent components seems arbitrary. No justification is provided for why this particular prior structure is appropriate for neural network weights in causal inference. The sensitivity to hyperparameters (λ_n, σ_0, σ_1) is not investigated.\n4. Assumption 2 (Overlap): Requires overlap on the imputed confounder Z, but how can this be verified when Z is unobserved?\n5. The paper does not provide clear, verifiable conditions to determine when causal effects are vs. aren't identifiable\n6. The relationship between \"loss-invariant transformations\" of confounders and causal effect identifiability needs more rigorous treatment\n7. What happens when Assumption 1(ii) is violated (i.e., true model is not a sparse StoNet)?\n8. All experiments use relatively small networks and simple settings\n9. Missing ablation studies on key components (e.g., impact of sparsity, SGHMC vs. standard SGD)\n10. Computational complexity not discussed. No analysis of computational cost, scalability, or convergence speed compared to baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)