MissNODAG: Differentiable Learning of Cyclic Causal Graphs from Incomplete Data
Authors: Muralikrishnna Guruswamy Sethuraman
Razieh Nabi
Faramarz Fekri
Venue: Accepted by TMLR ()

Abstract: Causal discovery in real-world systems, such as biological networks, is often complicated by feedback loops and incomplete data. Standard algorithms, which assume acyclic structures or fully observed data, struggle with these challenges. To address this gap, we propose MissNODAG, a differentiable framework for learning both the underlying cyclic causal graph and the missingness mechanism from partially observed data, including data *missing not at random*. Our framework integrates an additive noise model with an expectation-maximization procedure, alternating between imputing missing values and optimizing the observed data likelihood, to uncover both the cyclic structures and the missingness mechanism. We establish consistency guarantees under exact maximization of the score function in the large sample setting. Finally, we demonstrate the effectiveness of MissNODAG through synthetic experiments and an application to real-world gene perturbation data.

Published in Transactions on Machine Learning Research (04/2026)
MissNODAG: Differentiable Learning of Cyclic Causal Graphs from Incomplete Data
Muralikrishnna G. Sethuraman
mwwlliggs@ggtechh.oo edu
School of Electrical and Computer Engineering Georyia Institute of Technology
Razich Nabi
maich.nabiiermoo
Department of Biostatistics and Bioinformatics
Emory University
Faramarz Fekri
Yaramara.fedrr@cce.ggeeee
School of Electricol and Computer Engineering Georgia Institute of Technology
Reviewed on OpenReview: https: // opwnreview..eeeee.eo net/ net/forum forum? id- =NZXQ300GP
Abstract
Causal discovery in real-world systems, such as biological networks, is often complicated by feedback loops and incomplete data. Standard algorithms, which assume acyelic structures or fully observed data, struggle with these challenges. To address this gap, we propose MissNODAG, a diffcrentiable framework for learning both the underlying cyclic cansal graph and the missingness mechanism from partially observed data, including data missing not al random. Our framework integrates an additive noise model with an expectation ietrratiionmaimiizatiiinn procedure, alternating between imputing missing values and optimizing the observed data. likelihood, to uncover both the cyclic structures and the missingness mechanism. We establish consistency guarantees under exact maximization of the score function in the large sample setting. Finally, we demonstrate the effectiveness of Mis:OODA through synthetic experiments and an application to real-world gene perturbation data.
1 Introduction
Causal discovery, the process of identifying causal relationships from data, is fundamental across scientific domains such as biology, economics, and medicine (Spirtes al., 2000; Sachs al., 2005; Zhang al., 2013; Segal et al., 2005; Imbens & Rubin, 2015). Understanding these relationships is crucial for prodicting how systems respond to interventions, enabling informed decision-maken in complex systems (Solus al., 2017; Sulik et al., 201 7; Sethuraman et al., 2023). Traditionally, causal relationships are modeled using directed graphs, where nodes represent variables, and directed edges encode cause-effect relationships.
Existing causal discovery methods are typically divided into two main categories: constraint-baat based and score- based approaches. Constraint-based methods, such as the PC algorithm (Spirtes et al., 2000; Triantafillou Tsamardinos, 2015; Heinze-Deml al., 2018), infer the causal structure by enforcing conditional independencies observed in the data, though they often struggle with scalability due to the large number of required conditional independence tests. Score-based methods, such as the GES algorithm (Meek, 1997; Hauser & Bühlmann, 2012), optimize penalized score function, like the Bayesian Information Criterion, over the space of candidate graphs, usually employing greedy search lechniques. There also exist hybrid met hods which combine elements of both approaches, leveraging conditional independence tests alongside score optimization (Tsamardinos et al., 2006; Solus al., 2017; Wang et al., 2017). Recent advances have introduced differentiable discovery methods, such the NOTEARS algorithm (Zheng et al., 2018), which frames learning of Directed Acyclic Graph (DAG) as continuous optimization problem, enabling scalable and efficient solutions via. gradient-based

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methods. Following NOTEARS, several extensions have been developed for learning DAGs under various assumptions in observational settings (Yu et al., 2019; Ng et al., 2020; 2022; Zheng et al., 2020; Tee et al., 2019).
Most causal discovery methods make one or both of the following assumptions: (i) the data is fully observed, and (ii) the underlying graph is acyclic. However, real-world systems often violate these assumptions. Biological systems, such as gene regulatory networks, and socio-economic processes frequently exhibit feedback loops (cycles) (Sachs et al., 2005; Freimer et al., 2022), while missing data is common in practical applications (Getzen et al., 2023). These complexities significantly limit the applicability of standard causal discovery methods.
Missing data mechanisms are classified inlo three categories: Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR) (Little & Rubin, 2019). One common approach to dcaling with missing data involves discarding incomplete samples or excluding variables with missing data (Carter, 2006; Van den Broeck et al., 2015; Strobl et al., 2018), which might be suitable only in restrictive settings such as MCAR mechanisms while missingness rate is negligible. Otherwise, it leads to performance degradation as the missingness increases. Another common approach is imputation- based methods where the missing data is first imputed before applying causal discovery algorithm on the data. Some notable imputation algorithms include multivariate imputation by chained equations (MICE) (White et al., 2011), MissForest (Stekhoven & Bühlmann, 2012), optimal transport (Muzellec et al., 2020), and a few deep learning based approaches (Li et al., 2019; Luo et al., 2018). However, imputation-based methods typically assume that data are MAR, which can lead to bias when the data are actually MNAR, a common occurrence in practice (Singh, 1997; Wang et al., 2020; Kyono et al., 2021; Gao al., 2022). Gain & Shpitser (2018) addresses ANAR data by using reweighted observed cases as input to the PC algorithm alongside a weighted correlation matrix. Additionally, Tu et al. (2019) extends the PC algorithm by incorporating corrections to account for both MAR and certain cases of MNAR, while also learning the underlying missingness mechanisms.
Furthermore, while the acyclicity assumption simplifies computations by factorizing joint distributions into conditional densities, many real-world systems feature cyclic relationships (Sachs et al., 2005; Freimer el al., 2022). Several approaches have been developed to relax the acyclicity assumption, allowing for cyclic causal graphs. For example, carly work by Richardson (1996) extended the sestrraindd-bssedd framework to account for cycles, and Lacerda et al. (2008) provide an Independent Componenet Analysis (ICA) based causal discovery for linear non non-Gaussian cyclic graphs. More recent score- based methods for learning cyclic graphs include (Huetter & Rigollet, 2020; Améndola et al., 2020; Mooij & Heskes, 2013; Drton et al., 2019). Additionally, methods such as those proposed by Hyttinen et al. (2012) and Huetter & Rigollet (2020) focus on learning cyclic graphs from interventional data. Sethuraman et al. (2023) further extended this line of approach to nonlinear cyclic directed graphs, eliminating the need for augmented Lagrangian-base based solvers by directly modeling the data likelihood.
Contributions. In this work, we address two major limitations in causal discovery: the inability to handle informative MNAR data and the restriction to acyelic structurcs. We propose MissNODAG, general framework that extends expectation expectation-mmxiiiaailll maxinization (EM) based causal discovery to eyclic graphs and arbilrary graphically represented MNAR mechanisms. MisSNODAG alternates between imputing missing values and maximizing the expected log- likelihooho of the observed data, unifying and extending prior approaches such as MissDAG (Gao et. al., 2022). Following Sethuraman et al. (2023) and Behrmann et al. (2019), we employ residual normalizing flows to flexibly model data likelihoods in both linear and nonlinear structural equation models. Our framework accommodates MCAR, MAR, and MNAR missingness and can represent any MNAR process that admits a graphical specification. Through synthetic experiments, we show that NOssNODAG outperforms state-of-the-art imputation ion techniques combined with causal learning on partially missing interventional data. Furthermore, we also provide consistency guarantees for exact maximization of the score function.
The paper is organized follows. In section 2, we describe the problem setup and outline the modeling assumptions. Section 3 introduces the proposed popectatoon-mxximizaton---aee MissNODAG framework. In section 4, we validate MissNODAG on various synthetic datasets. Section 5 concludes the paper. All proofs are deferred to the appendix.
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2 Problem Setup
A list of notations is provided in Appendix A for the ease of reference.
Structural Causal Model. Let 9 = (X denote a possibly cyclic causal graph with a set of vertices
X = (X1,..., Xk), representing a vector of K random variables connected by directed edges & S X x X. We assume the following structural causal model (SCM), also known as structural equation model (SEM), with additive noise terms to capture the functional relationships between variables in g (Bollen, 1989; Pearl, 2009):
X f&r (pa)(Xx)) + ck. k k= K,
(1)
where (X²( =
+ X2 E E} denotes the parents of X in G. The function fx describes the relationship between Xę and its parents, with Gg as the exogenous noise term, assumed to be mutually independent (no unmeasured confounders), and collected as = (f1...., ck). We assume that self-loops (edges of the form X + Xx) are absent in ((X), as this could lead to model identifiability issues (Hyttinen et al., 2012).
Let F(X) collect the functions (x(pag(X))) for all k. The structural equations in eq. (1) can be written as follows:
x F
(2)
Let id denote the identity map, so the forward map (id F) maps X to e. We assume that this mapping is bijective, ensuring the existence of (id- - =F)- (called the reverse map), and that both (id iid-Py P) and (id - F)- are differentiable. The former ensures that there is a unique X for a given 6, thus, we can express X as X= (id - Fc). This assumption is nceded for our dovelopments in section 3.3, and is naturally satisfiod when the underlying graph is acyelic (Mooij & Heskes, 2013; Sethuraman el al., 2023).
Intervention operations can be readily incorporated into eq. (2). In this work, we focus exclusively surgical, or hard, interventions (Spirtes et al., 2000; Pearl, 2009). Graphically, a hard intervention corresponds to removing all incoming edges to the intervened variable. Following similar notational convention in (Hyttinen et al., 2012; Sethuraman et al., 2023), given a set of interventional targets Xz, we can decompose X into disjoint sets, X Xx U Xo, where XI represents the set of intervened variables in an interventional experiment, and Xo represents the set of purely observed variables. We denote the graph obtained after performing the interventions Xz as (9). Let D {0, 1) be diagonal matrix where DEs 1 if X2 E Xo, and 0 otherwise. Under this setting, the SEM in eq. (2) is now modified to:
X DF DE +C,
(3)
where C denotes a vector of size K representing intervention assignments for variables in X. Specifically, Ck = Xx if Xg Xz, and C = 0 otherwise. Let co denote the exogenous noise terms corresponding variables in Xo. Let Peo (EO) and pxz (Xx) be the joint probability densities of eo and Xz, respectively. We thus have,
Px xx = (Xz) (co) |det (X))
(4)
where det ((aa-Dj)))) (X) denotes the determinant of the vector-valued Jacobian matrix of the function (id-DF) at X. See proof in appendix B.1.
Missing Data Model. Given sampled data on X, let R = (RI...., Rx) be the vector of binary missingness indicators with R& = 1 if X₁, is observed and R& = 0 if X is missing. We only observe coarsened version of Xk. denoted by Ys, which is deterministically defined as Y, = X2 when R3 = 1, and Y3- = if R₂ = 0. Let Y = -Yh,..., YR) denote the coarsened variables. Additionally, we have access to = (S,,..., Sk) where Sk a binary indicator of intervention, such that St = 0 if X3 is intervened on (i.c., X₁ E Xz), and Sa- = otherwise. We assume we have n i.i.d. copies of (Y, R, S), and the dataset denoted by (()
D (,)), (0), 500 J where ((6), ,(0), s(() represent the i-th observed values of Y, S.
We define a missing data model as the collection of distributions over the variables N.R.Y. By chain rule of probabilities, we can express p(X, Y) as low, X( as the missingness mechanism, p(X, R) as the full law, while p(Y[X, R) is the coarsening mechanism, which is deterministically defined. Borrowing idcas from the graphical models of missing data (Mohan et al,, 2013; Nabi al., 2025), we use graphs to encode assumptions about P..
We refer to p(X) as the largel
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Specifically, assume that the relations between variables in the target law p(X) are directed and can include cycles, and the missingness mecha- nism p( X)) factorizes according to a DAG, where pag(Rx) (Rx) can only be a subset of X and \ Rg- Finally, due deterministic relations, Y& has only two parents R3 and XR. We denote these graphs by 9m (V, Em], where V (X, R. Y), and &m denotes the set of directed edges in the missing data graph. Two examples of missing data graphs (or m-graphs), with K = 3 substantive variables, are provided in figure 1; deterministic relations are drawn in gray.
X1
X3
X1
X2
X3
Ri
R2
R3
Ri R2
R3
11
11
1/2 Y2
Y.2
(a)
(b)
Figure 1: Example m-graphs with three variables illus-
trating: (a) An example an MNAR mechanism where no odge of the form R3 + RG exists; (b) An MNAR mechanism where I's are connected and the full law is identifiable (no self-censoring and colluder structures).
Graphically, MCAR mechanism has incom- ing edges into any missingness indicator in R, MAR mechanism has parents of missingness indicators that are fully observed, and an MNAR mechanism involves missingness indicators with parents in X. Identifying the full or target law in an m-graph with MNAR mechanisms from observational data is not always possible. Previous work has extensively studied identification in graphical models of missing data (Bhattacharya et al., 2020; Nabi et al., 2020; Mohan Pearl, 2021; Guo et al., 2023; Nabi et al., 2025; Guo & Nabi, 2026). Nabi al. (2020) have shown that the full law in m-graph is identified and only there are edges of the form X₁ Rg (no self-censoring) and X; R& Rj (no colluders). Therefore, restrict the m-graphs considcred in this work to be identifiable (formalized in assumption 1).
Assumption 1. The missing data graphs under consideration have edges of the form X - Ra (no celf-censoring) and X, - R2 + R2 (no colluders).
Given partially observed data from set of interventional experiments, our objective is to learn the underlying full law that generated the sample. Specifically, this involves learning both the undorlying target law and the missingness mechanism.
3 The MissNODAG Framework
We assume the target law p(X) and the missingness mechanism p(R[X) are parameterized by finite vectors 0 and Φ, respectively. Thus, we write the full law (X, R) as p(X, R(O, Φ) = p(X(0) p(R|X, the full law, we proceed by maximizing the log-likelihood of the observed data law.
(R) Φ). In order to learn
Let I, = {k: ( = 1} and Q4 = {k: r = 0} represent the sets of indices for the variables that are observed and missing, respectively, in the i-th sample; thus (9)) oT. can be written down
(
Consequently, the observed data law p
(3) (())
p( (0) 0. 10,0 S0af///faaao 2 2 (0,0 (0) 1,010,00 10,0) d
5
This integration is generally intractable due to marginalization over missing variables and the lack of a closed-form solution. We address this intractability in maximizing the observed data likelihood when data is generated from an m-graph, as described in section 2. First, we describe the score function to be maximized and establish the consistency result section 3.1. Then, we provide an overview of the MissNODAG framework in section 3.2, followed by details on computing the log-likelihood for missing and observed variables, and discuss imputing the missing variables under linear and nonlinear SEMs in section 3.3.
3.1 Score Function
Assuming the data D (,) (1), (()),,())))) 500) 1 is generated from an experiment where the full law p(X, R) is represented via m-graphs, our goal learn the entire m-graph structure by maximizing, L(D, 0, Φ),
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regularized oigllikel of the observed data law:
max log ( 0, の) - (,R(0)- (22((0) s.t. 11(0) = and 12(()) = 0,
(6)
==1
where R(-) is a regularization function that promotes sparsity, \ regularization parameters, and the functions hi (6) and (0)) are functions that restrict the missingness mechanism to the class of identifiable MNAR mechanisms, see section 3.3.3 for more details.
1M 1m under the population (infinite data) Given a family M interventional experiments with targets (m setting, the regularized observed data likelihood score for an m-graph 9m, denoted as S(gm) given by
M Σ
S(gm) = sup
E
(m) ((g,((r, R) 11
(7)
Ogm PP WEL
where PoDs is the true data generating distribution for the m-th interventional setting, |mm| combines R(0) and R(Ø) and denoles sparsity regularization on the m-graph (number of edges), and À denotes the combined sparsity regularization parameter. The following theorem establishes that, under standard assumptions assumptions, the exact maximization of S(G) results in a graph 9m that is in the same Markov equivalence class as the ground truth graph 9 Formally,
Theorem 2. Let { Jm- be a family of interventional largels, let I denole the ground trulh directed mixed graph, plm) denote the data generating distribution for XIm, and 9m == arg maNg ED S(Gm), where Gia denotes the set of all m-graph that obey assumption 1. Then, under assumptions 1, B.7 to B.9 and B.11, and for a suitably chosen x > 0, 9m is I-Murkov equivalent to Gin, denoled 9m =7 9
The proof for theorem 2 is provided in appendix B.3. This result exlends the consistency guarantee established by Sethuraman & Fekri (2025) under complete data to the missing data setting. While exact maximization of the score function theoretically guarantees recovery of the ground truth Markov equivalence class, it is computationally infeasible in practice. Consequently, we resort to an Expectation-Maxxiizzttii algorithm to optimize the score function (detailed in section 3.2). Although EM does not strictly maximize the observed data likelihood liimttnng convergence to stationary point MissNODAG AAGi is capable of recovering the true target law graph when provided with all single-node interventions (see section 4).
(EM)
experiments indicate that
3.2
The Overall Procedure
As discussed, computing p(=f,, 1010 Φ) is generally intractable, with no closed-form solution. However, eq. (6) can be solved using the iterative penalized expectation maximization (EM) method (Chen et al., 2014). Unlike imputation methods that directly sample missing values, the EM algorithm starts with an initial parameter O (0%,g) Φd) and alternates between the following two sleps at iteration until convergence:
popectation-maxiiizzaioon
E-step: Use the current estimates of the model parameters, @t (0', Φ²), and the non-missing data compute the expected log- likelihoohd of the full data, denoted by Q(e)0*), and given by:
2 0*)
Σ
E 2n la
fo m0 o*) log (3) (0) [0)]
(8)
M-step: Maximize 2(0)0*), computed in the E-step, with respect θ:
0 = arg max Q( 0²) - AI -2)))) s.t. (i( -0 and h2(#) = 0.
(9)
We use stochastic gradient- based solvers solve the maximization problem, alternating between updating the parameters of the target law, 0, and the parameters of the missingness mechanism, Φ. Note that,
logp(a (6) (()) ()) N Q(e (+) - const.
(10)

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This incquality indicates that we maximize lower bound of the log-likelihood as update the parameters during the M-step. More details on the derivation of eq. (10) and on the convergence analysis of MissNODAG are provided in appendix B.2.
3.3 Computational Details of the E-step
Computing the expected log-likelihood in the E-step can be challenging for a directed (cyclic) graph. First, let's look at log p( T, D2 (0) in eq. (8), which equals:
(0)
logpx (ox(1f))) ( (T, Tn 10 + logp(r (3) (6) (2,
+zg "T. Ø).
(11)
3.3.1
Target Law
As per eq. (4), computing (ogpx(XX)) in eq. (11) requires:
log det ((a-DE)
(12)
In the worst case, the Jacobian matrix may require gradient calls in the order of K². To that end, following Sethuraman et al. (2023), we employ contractive residual flows (Behrmann et al., 2019; Chen et al., 2019) to compute the log-determinant of the Jacobian in a tractable manner.
Modeling the SEM in eq- (2). We assume the SEM functions in F(X) in eq. (2) are Lipschitz with Lipschitz constant less than one. Such functions called contractive functions. It then follows from Banach fized point theorem (Rudin, 1953) that the mapping function (id DF') is contractive and invertible.
We use neural networks to model the contractive functions in F(X). As shown by Behrmann et al. (2019), neural networks can be constrained to be contractive during the training phase by rescaling the layer weights by their corresponding spectral norm. While contractivity is a sufficient condition for the existence of (id - F) it is not necessary. When the underlying graph governing the target law p(X) is a DAG, is possible to have non-contractive functions in F(X) for which (id - FF)- exists; see (Sethuraman et al., 2023) for more details.
Naive implementation of a neural network may not produce promising results as it may introduce self-cycles. To circumvent this issue and simultaneously add sparsity penalization, we introduce a dependency mask matrix M ~ (0, 1) to explicitly encode the dependencies between the nodes in the graph, with zero diagonal entries to mask out the self-loops. Thus the SEM model F(X) takes the following form
F (X)] = (M,, O X)]s,
(13)
where NN, denotes fully connected neural network function with parameters 8, Ma,k denotes the k-th
column of M, and @ denotes the Hadamard product. The dependency mask is sampled from Gumbel-soflnax distribution (Jang et al., 2016), M ~ p(M (0) and the parameters 0 are updated during the training (M-step). In this case, the sparsity penalty R() in eq. (6) is set as an LI norm, i.c., R(0) = E EM [|
M|
Computing the log-determinant in eq. (12). We note that log det Jea- DRY (X) log det(I- (X)) where I is the K x K identity matrix. Thus, the log-determinant of the Jacobian matrix can be computed using an unbiased estimator based on the power series expansion introduced by Behrmann et al. (2019),
log det J(d- = log 11-JA JDP
Σ
Tr [T(
(14)
where Jmp (X) denotes the Jacobian matrix to the m-th power. Equation (14) is guaranteed to converge when F(X) is contractive (Hall, 2013). In practice, eq. (14) is computed by truncating the number of terms in the summation to a finite number. This, however, introduces bias in estimating the log-dcterminant of the Jacobian. In order to circumvent this issue we follow the steps taken by Chen et al. (2019). The power series expansion is truncated at random cut-off N PNs(N), where pn is a probability distribution over the set of
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Algorithm 1 IMPUTE-REJECTIO
Require: Minibatch data B {u(0) ,,(0 s(0)
p( (0) 16 10'))())))) 110000010000 (
oo, Mr, .d9)
4: u
then
with sampling distribution g(x).
(og(
Ensure: Imputed data B = {x(1), (1), 8(00)
11=1"
5: Accept sample: Til
(1)
yr.
(8)
1: for i = 1 to na do
end if
2: Sample 210 g(
end for
Pick u 00 1].
return B= (3 s() pam
natural numbers N. Each term in the finite power series is then re-weighted to obtain the following estimator:
Tr Tr(JBp(X) Px(e m)
log det J(a J(a-Dr)(X)) = -EN
(15)
where Pi is the cumulative density function of ps. Gradient calls are still required in the order of K. We can reduce this further by using the Hutchinson trace estimator (Hutchinson, 1989), where W ~ (N,
Tr(3Bo(x) Ew W JWJJ (X)W). (X)W]
(16)
We note that the remainder of eq. (4) can be efficiently computed with forward pass of the neural network.
3.3.2 Calculating Expectation via Imputation Combining eqs. (4), (8), (15) and (16), arrive at
N W WTJ (2 m PM(e 2 m)
@ x
E e0 20
log (100)
TT, (2) (() .f) .)) + Peo Peo (6).)
where (0) (id D2 (zx D is the diagonal matrix corresponding the interventional mask for the i-th sample, i.., D. diag(s) (1 (i) R)
The expectation in the approximation for Q(0(0*) generally lacks a closed-form solution. Therefore, it must (2) (+)
be approxinated by the sample mean, using samples drawn from the posterior distribution p(* [TT (0) O*). This presents two main challenges: (i) the posterior distribution may not have closed form, and (ii) direct sampling may be infeasible even when the posterior distribution can be evaluated. The difficulty arises due to the presence of nonlinear relations in F(X) and the missingness mechanism p(r(0)22f [ IT, o²), which may preclude straightforward sampling. Therefore, we employ rejection sampling (Koller & Friedman, 2009) to draw samples from a proposal distribution ((=(()). from which samples can be readily generated.
(t) (
To that end, constant co is chosen such that coq(x stated carlier the posterior distribution is not readily available. Thus, from Bayes rule, we have
(3) 2 p(a |ef . (0), O') for all I,..., . n. However,
(0) (1 (0. (1) (,)) pl rT. 100) pla lof. r
platf) (0))0))
where the denominator p(=f)) (0))))))))))))))))))))) (0)) can be evaluated using fully observed data. The first term in the numerator can be computed efficiently, as discussed in section 3.3.1. The second term is addressed in section 3.3.3. Before that, we explain how these calculations simplify under more restrictive models. The imputation procodure is summarized algorithm 1. In our implementation, is possible that one pass of rejection sampling may not yield enough samples. Thus, we repeat the rejection sampling procedure until we recover at least half of the batch.

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3.3.3 Missingness Mechanism
In order to compute the log- ligeliiel in the E-step, we also need to compute log (ogp(r(0)1)) (2) according to m-graphs can be factorized as
12 "T, 0), which
p(R| 0) = II plp((t pag... (Bx), Φx).
(17)
In developing MissNODAG, we focus on a class of MNAR models, where for any RS E R. pag.. (Rs) S XU R Xx. Rg. According to the results in Nabi et al. (2020), the full law under any MNAR mechanism that follows assumption 1 (i.e, no self-censoring edges and no colluder structures) is identified as a function of the observed data distribution. Figure 1 provides two examples of identifiable MNAR mechanisms with K = 3 variables.
Under this MNAR class, cach conditional factor in the missingness selection mechanism pl p[][X, Φ) is modeled using the expit function, with ΦE = { z
p( RR = 01 p.g (R&), 0n) expit (xpi + (Wnn) (W =t):
(18)
where expit(x) 1/(1+ WXR and WR denote the adjacency matrix corresponding to the edges between X to R and R to R respectively. Maximization during the M-step with respect to ø reduces to solving constrained sparsity-regnlarized logistic regression, with our choice of R(Ø) Σx= ||wx||1-
Identifiable MNAR mechanisms are particularly well-suited for modeling missingness in cross-sectional studies, surveys, and retrospective analyses. Consider, for instance, a study examining the relationship between smoking (X1), tar accumulation in the lungs (X2), and a bronchitis diagnosis (X3), where missing entries are indicated by R2, R2, and R3, respectively (see figure 1). The edge X3 + R2 reflects that a suspected diagnosis of bronchitis increases the likelihood of inquiring about smoking history. Smokers are more likely lo undergo tests for both tar accumulation and bronchitis, represented by R2 X1 + R3. Moreover, requesting diagnostic test for bronchitis increases the likelihood of testing for tar, which in turn makes it more likely that smoking status will be queried captured by Rí + Ra +- R3 in figure 1(b); while in the absence of such a dependency, figure 1(a) is most suitable to illustrate the missingness mechanism.
3.4 Enforcement of MNAR Identifiability Constraint in the M-step
To ensure that the model class adheres to identifiable MNAR missingness mechanisms, explicitly incorporate the identifiability conditions (assumption 1) into the optimization objective. The first condition, no self- censoring, can be enforced by masking the diagonal entries of WXR. The second condition, no colluders, requires more care: we must avoid structures of the form X₂ + R, + R4- Here, the edge X₃ + R corresponds to (WxR)R.j, and Ry- < Rg to (WRR)R.. To preclude both edges from co-occurring, we require (WxR)kJ - (WRR)aJ = 0 for all k.j [K]. Define h1(0) = x @ W WRRll1. where @ denotes the Hadamard product. Then enforcing hy (+) 0 guarantees the no-colluder condition, as formalized below:
Proposition 3. Setting h (+) = O is equivalent to requiring (WxR)es (WRR)as =0 for all k.j E [K].
Proof. This equivalence follows directly by expanding the f--norm in hi Note that,
hi(0) |XR WRRllI
Σ ((XR)) (WRR)asl- k.j=1
(19)
Setting h to zero in eq. (19) implies that for each k,j [K], (WxR)&.J" (W 0.
Beyond the identifiability conditions, we require the missingness indicators to lie outside any cycles that is, the graph encoding p(R] | X) must form a dirccted acyclic graph (DAG). Our paramoterization of the missingness mechanism precludes edges of the form R, + X,, so it suffices constrain WAR RR to the space of DAGs. This can be achieved using the trace-exponential acyelicity constraint proposed in Zheng et al. (2018),
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namely, (2( = parameters are updated by solving the following constrained optimization problem:
K = 0. In summary, during the M-step, the missingness mechanism
min @
ΣE
E e "n -p (o, ,0,00) [ogg (110) Ir. 2n +]
subject to
WxR O WRRl = and Tr(e WeeWnnGWag - K
bi(0): colluders constraint
ha(0): DAG constraint
Parameter update via Augmented Lagrangian. Following Zheng et al. (2018), we use the augmented Lagrangian method to solve the constrained optimization problem above. This approach augments the original objective with quadratic penalty, yielding:
min
Σ E 1=1
(0) 11 ,0,, [logp (r(0) IT. 2a 20 +
hz(d)
subject to (1) = 0, and h2 0,
where p > O is the penalty coefficient. A key advantage of the augmented Lagrangian approach is that it approximates the solution to the constrained problem without requiring to tend to infinity. This allows us to reformulate the objective as an unconstrained problem:
2 Σ h2( Σ xgh((
min
Σ
E e in
(0, ,0,))) logp (n(0) of 2
)]
(20)
where d and d2 are the dual variables associated with the constraints. These are updated at each iteration via:
A A ph:(o,,) i= 1.2.
(21)
4 Experiments
Using both synthetic data and a real-world gene perturbation dataset, we compared MissNODAG" to MissDAG (missdag) (Gao et al., 2022), an EM-based causal discovery method limited to DAGs and MAR missingness, MVPC (Tu et al., 2019) constraint-based method capable of handling MNAR missing data. Since MVPC does not handle interventions out-of-the-box, we apply the joint causal inference (JCT) (Moojj et al., 2020) framework along with MVPC to learn from interventional partially observed data. We also tested state-of-the-art imputation methods, including optimal transport (optransport) (Muzellec et al., 2020), MissForest (missforest) (Stekhoven & Bühlmann, 2012), and mean imputation, followed by causal graph learning from the imputed data using NODAGS-Flos (Sethuraman et al., 2023). Additionally, we also compare NOsSNODAG with ENCO (Lippe et al., 2022) after imputing the missing data with optransport imputation method, see appendix for details on the implementation of MissNODAG and the baselines. Note that, ENCO, MissDAG, and MVPC are only capable of learning DAGs, putting these methods at a disadvantage compared to the other baselines. Moreover, MissDAG doesn't 't handle MNAR missingness.
4.1 Synthetic Experiments
In all experiments, we generated cyclic directed graphs with K=10 nodes using the Frdös-Rény' (ER) random graph model. For linear SEMs, edge weights were uniformly sampled from (-0.6.00.25) U (0.25, 0.6) and rescaled to ensure contractivity. For nonlinear SEMs, the causal mechanism was defined as F(X)
tanh(W" TTX,, where the nonzero entries of W were drawn from the same distribution and similarly rescaled for contractivity. Exogenous noise variables were sampled from a Gaussian distribution with standard deviations randomly chosen between 0.1 and 0.3. The m-graph was generated using the ER model, followed by postproccssing to enforce acyclicity in W WRR- Missingness mechanism edge weights were sampled from
The code for MissNODAG available https: http:://ggttt...m//aaaagggggggge cww.m/maaallliigggiiieenonag

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Published in Transactions on Machine Learning Research (04/2026)
Ablation study (varying missing probability)
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
SHID
SHD
10
0.2
0.3
0.4
0.5
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0.5
Av. missing prob
Av. missing prob
Av. missing prob
Ablation study (varying degree of nonlinearity)
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
10
SHD
SHD
SHD
10
5
0.25
0.5
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0
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Degree nonlinearity (0)
Degree nonlinearity (0)
Ablation study (varying cycles)
Target Law
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Missingness Machanism (R2R)
30
10
SHD
SHD
SHD
10
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Num. cycles
Num. cycles
Ablation study (varying number of training interventions)
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
10
SHID
SHI
Num. Interventions
Num. Interventions
Num. interventions
MisNNODAG MisDAG
Missforest
Op-Transport
Mean ENCO - MVPC + Logistic
Figure 2: Performance comparison on synthetic data
a standard normal distribution. The training data comprised single-node intorventional experiments for all nodes in the target graph (unless specified otherwise). Upon creation of the clean data, some of the observations are replaced with s?" (indicating missing value) based on the missingness mechanism modeled by the m-graph. We evaluated MissNODAG and baseline methods across the following five settings.
Sensitivity to missing probability. The average probability of node being missing (E[R,)) varied between 0.1 and 0.5. Each interventional setting consists of 1000 samples. Additionally, the outgoing edge densities of the target law and the missingness mechanism set to 2 and 3 respectively.
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Ablation study (varying number of nodes)
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
80
60
30
S1DD
S11D
40
SHD
40
20
10
20
10
20
30
30
10
50
Num. nodes
Num. nodes
Num. nodes
MisNNODAG - MissDAG Miesforest + Op-Transport - Mean
ENCO + MVPC +
Logistic
Figure 3: Performance comparison between MisNNODAG and the baselines as function of the number of nodes in the target law graph.
Sensitivity to nonlinearity.
We varied the degree of nonlinearity by adjusting 3 e [0, in:
X (1- X tanh(W (x)x))+ +E.
(22)
Here, B = 0 corresponds to fully linear SEM, and B the target law graph K was set to 10, missing probability set to 0.3, and the number of cycles were randomly chosen.
fully nonlinear SEM. The number of nodes in
Sensitivity to cycles. We varied the number of cycles in the target-law graph from 0 to 8, fixing the number of nodes at K = 10 and setting 9 = in eq. (22). The missing probability was set to 0.3.
Sensitivity to training interventions. We varied the number of interventions provided during the training phase. Starting from 0 interventional experiments in the training set (corresponding to observational setting) to all single-node interventional experiments (10 interventions). Here K was set to 10, missing probability to 0.3, 3 1.0 in eq. (22).
Sensitivity to number of nodes. In this case, the number of nodes in the target law graph was varied between and 30, with the number of cycles assigned random and B 1 eq. (22). The missing probability was set to 0.3.
Structural Hamming distance (SHD) is used as the error metric to compare MisGNODAG with the baselines. SHD counts the number of operations (addition, delction, and reversal) needed to match the estimated graph to the ground truth.
Target law recovery. We first analyze target law recovery performance, sunmarized in the left columns of figures 2 and 3. Across all five settings, MissNODAG either outperforms or matches the baselines. In the experiment with varying missingness probabilities (figure 2, top row), the performance of all methods declines slightly as the average missingness probability increascs; however, MissNODAG achieves nearly perfect recovery up to missing probability of 0.3. In the second and third settings (varying nonlinearity and cycle counts), MissNODAG remains robust, achieving SHD scores close to 0. regarding the number of training interventions, MissNODAG performs competitively; while ENCO performs better in the purely observational setting (zero interventions), MissNODAG overtakes it as the number of interventions increases, widening the performance gap. Finally, regarding scalability (figure 3), while all hods deteriorate as the graph size grows, MissNODAG remains the top-performing model for graphs up to K = 40 nodes.
Missingness mechanism recovery. For analyzing the missingness mechanism recovery, the baseline model applics logistic regression, treating cach missingness indicator R as the target and using the remaining indicators R_ along with X as features. We evaluate recovery in two parts: X R edges and R + R edges. For the latter, we convert the estimated graph into its completed partially directed acyclic graph
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(CPDAG) and compute SHD against the ground truth CPDAG. Results for X + R edge recovery are shown in the middle column of figures 2 and 3, and for R + R edge recovery in the right column of figures 2 and 3, with baseline models denoted as logistic. On X + R recovery, MissNODAG consistently outperforms the baselines across all five settings, with a more pronounced performance gap than in target law recovery. For R - R recovery, MissNODAG again matches or exceeds baseline performance in all settings, though the difference is less pronouneed.
4.2 Real-World Experiment
Here we present an experiment focused on learning causal graph structure corresponding to gene regulator network from single-cell RNA-sequencing (scRNA-seq) dataset with genetic interventions. In particular, we focus on the Perturb-CTTE-see dataset (Frangieh et al., 2021), a type of data set that allows one to study causal relations in gene networks at an unprecedented scale. It contains gene expressions taken from 218,331 melanoma cells split into three cell conditions: (i) control (37,627 cells), (ii) co-culture (73,114 cells), and (iii) interferon (INF)-y (87,590 cells).
Due to practical and computational constraints, we restrict our analysis to a subset of 10 genes (out of 20,000), following the experimental setup of Sethuraman et al. (2023), as summarized in table 1. We include all single-gene interventions targeting this subset, treating each cell condition as independent dataset on which models are trained separately.
A key challenge in scRNA-see data is the prevalence of zeros, a phenomenon known as dropout. While some zeros reflect true biological absence of gene expression (Zappia al., 2017), others are technical artifacts introduced by the sequencing process (Jiang et al., 2022; Ding et al., 2020). In our work, treat all zero expression values missing and apply MissNODAG to impute them during training. Although this approach may misclassify genuine biological zeros as missing, we find that treating zeros as missing improves model performance compared to assuming fully observed data.
Table 1: List of genes chosen from Perturb- CITE-seq dataset (Frangieh el al., 2021).
STATI SSR2 DNMTI
B2M
LGALS3 TM4SF1
PTMA
CTPS1
MRPL47
TMED10
Since the data set does not provide a ground truth causal graph, it is not possible to directly compare the performance using SHD. Instead, we compare the performance of the causal discovery methods based on its predictive performance over unseen interventions. To that end, we perform 90-10 split on the three data sets. The smaller set is treated as the test sel, which is then used for performance comparison between MissNODAGS and the baselines. As a performance metric, we use the predicted negative lig-likeld (NLL) over the test set after training the models for 100 epochs. The results are summarized in figure 4.
MissNODAG out perforosss baseline methods across all three cell conditions. Furthermore, MissNODAG and other inputation-based approaches (mean and optransport) consistently surpass NODAGS- directly on unprocessed data. The middle plot of figure presents sensitivity analysis in which we vary the proportion of zero gene concentrations treated as missing data (dropouts) from 0% to 100%. The results indicate that predictive performance improves as a larger fraction of zero values is explicitly modeled as dropout. These findings highlight the prevalence of dropouts in AcR.NA-se datasets and underscore the critical importance of accounting for them when modeling gene expression data.
trained
Finally, we provide the estimated gene regulatory network in figure (right). NOsANODAG has successfully recovered key regulatory structures and feedback loops consistent with established T-cell biology. The directed edge STATI + B2M correctly identifies primary regulatory mechanism where the transcription factor STATT drives the expression of antigen presentation machinery, well-known response to immune signaling (Frangieh et al., 2021). The hub of connections surrounding CTPS1 validates the model's ability to detect functional metabolic modules, as CTPS1 is the rate-limiting enzyme specifically required for T-cell proliferation ion and DNA synthesis (Martin et al., 2014). The detected cycle between DNMT1 and STATI captures a sophisticated epigenetic feodback loop, where DNMT1 required for cell division but simultancously acts to repress STAT1 to prevent excessive inflammation, validating the method's capacity to identify cyclic causal dependencies in biological data (Zhang et al., 2005).
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Performance Perfurb-CITE-seq
Sensitivity Analysis Missing
Estimated Causal Graph
1.6
MisNODAG
Op-Traeprrts NODAGS
1.4
Co-culture Contzol
BBM
1.4
IFN-7
DXSTI
LGALS3
1-NLL
1.2
I-NLL
MRPL4
ВЕXT1
Control
09%
25%
30%
100%
condition
Portion of zeros missings (%)
Figure 4: Evaluation on BerturbCCTT--eey gene perturbation data. (Teft) Performance comparison the Perturb-CITE-ser ceRNA-seq dataset with d = 10 genes, measured using interventional negative log-like log-likelihood (I-NLL). Zero gene expression values are treated as missing, (Middle) Sensitivity analysis for MissNODAG on the different cell conditions with respect to the fraction of aero-valuee gene expressions treated as missing; 0% corresponds the fully obscrvod sctting (NODAGS-Flow). (Right) Adjacency matrix the estimated causal graph for the co-culture cell condition, where red entries indicate the presence of directed edge from the gene corresponding to the row to the gene corresponding to the column.
5 DISCUSSION
In this work, we proposed NOssN novel differentiable causal discovery framework capable of learning nonlincar cyclic causal graphs along with the missingness mechanism from incomplete interventional data. The framework employs an values and optimizing model parameters. We demonstrated how imputation can be efficiently achieved using rejection sampling, and in the case of linear SEMs with MAR missingness, by directly sampling from the posterior distribution. One of the key strengths of MissNODAG is its ability handle cyclic directed graphs and MNAR missingness, significant. advancement over methods that typically focus on DAGs and MAR mechanisms.
maximization algorithm that alternates between imputing missing
MissNODAG has three primary limitations. First, our theoretical guarantees rely on strict identifiability assumptions (assumption 1); as shown in appendix C.2, violating these conditions degrades performance, particularly for the recovery of missingness mochanism. Second, the non-convex optimization landscape means the EM procedure guaranlees convergence only to stationary point, not global maxinum. Finally, the log-determinant of the Jacobian computation imposes computational complexity of O(K)L), limiting scalability to large-scale graphs
Future research directions include: (1) incorporating realistic measurement noise models in the SEMs to enhance robustness in real-world datasets, as explored in linear DAG models by Saced et al. (2020): (2) scaling the current framework to larger graphs using low-rank models and variational inference techniques, as explored in acyelic structures by Lopez al. (2022); (3) allowing for unobserved confounders within the modeling assumptions, as was explored under complete observations by Bhattacharya et al. (2021) and Sethuraman & Fekri (2025); and (4) generalizing models, while (ackling the challenges of non-identiffabilit in the full or targel laws, explored in DAG models by Nabi & Bhattacharya (2023) and Guo et al. (2023).
framework to broader classes of identifiable MNAR
Acknowledgments
This material is based work supported National Science Foundation (NSF) under grant number 2502298.
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Appendix
The appendix is structured as follows. Appendix A offers a summary of the notations used throughout the manuscript for ease of reference. Appendix contains all the proofs. Appendix C provides details on three sets of additional simulations and a comparison of training times. Appendix provides additional details on the implementation of the baselines and the MissNODAG framework.
A GLOSSARY
A comprehensive list of notations used the manuscript. provided in Table A.I.
Table A.1: Glossary of terms and notations
Symbol
Definition
Symbol
Definition
X,
Variables, values
g, 9m
Graph, m-graph
Exogenous noise terms
Graph node index variable
R
Missingness indicators
K
Total nodes in
Y
Coarsened version X Intervention indicalor vector Set of intervened nodes Set of non- inn-intereeeed nodes Exogenous noise density function Target, full laws Coarsening mechanism
(ag(X))
Parent set of X2 in g
S
F, Sk
SEM functions
Xz
id
Identity map
Xo
(id F)
Function mapping X to
pe(e)
(x(X)
Jacobian matrix of F at X
P(X,)
K K Identity matrix Matrix of intervention masks
P(Y|X, R)
D
p(R[X)
Missingness mechanism
B
Weighted adjacency matrix of lin- ear SEM
P(M
Gumbel-Softmax distribution for sampling M
M
Dependency mask for SEM func- lion F
θ = ())
Parameters of px (X)),, X)
9 = 92
Interventional Markov equivalence Intervention mask corresponding to i-th sample
O
Parameters at EM t-th iteration
D
Index sets for observed, missing nodes
i
Sample index variable
Wk.
Parameters of p(Rk pagm (Rx))
Total sample size Mini batch size in each EM
(0) (0)
i-th sample
A B
Hadamard product of A and B
Power series expansion index for log-determinant of Jacobian
Il
L1 norm
N
Number of power series terms
((-)
Proposal distribution for rejection sampling
W
Gaussian random variable used for computing trace of Jacobian
R(-)
Sparsity inducing regularizer
Matrix exponent of A DAG constraint. function
(1)))
No colluders raistrain function
h2(#)
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B Theory
B.1 Joint Density of Target Law Under Intervention
Consider the structural equation model X = F(X)+ tc, which implics X (id-F) ((d-F)-((d)) (c). Using the properties of probability density functions, the joint distribution of X = -X Xx) is given by,
(x = Pc ((id - FF)) det J (x)
(23)
where (e) is the probability density function of the exogenous noise vector Under an interventional setting Xx), all incoming edges to the nodes in X7 are removed, leading to the following structural equations:
{ Xx {
X E X
X2
(A((pa(X)))
if Xx & Xx
That is, Xx is set to known value Xx if it is intervened upon, and the structural equation remains unchanged when X2 is purely observed. The above equation can be written more conciscly as follows:
Xx = dx (Pag + ex) + Cs. for k = -....,,.. K.
(24)
where dj 1[Xx & Xz), and 1.- is the indicator function, and Ck Xx if X E XI, and Cs otherwise. Let D R be diagonal matrix such that DRA du- Thus, eq. (24) can now be combined for k= -.....: K to obtain the following equation,
X x=DF(X)+Dc+C.
(25)
where F(X) is defined in section 2 and C (Ci...., Ck). Thus we have, (id- (X) = De + C, this implics that X = (id DF) (De + C). Let X2 px, (Xz). By the change of variable rule in probability, can write eq. (23)
(x( pxz (Xz) Peo (co)
det J(a-
(26)
where co is the exogenous noise terms of variables in Xo.
B.2 Convergence Analysis of the EM Algorithm
Here provide the convergence analysis of MissNODAG. Our analysis relies the convergence of the EM algorithm (Wu, 1983; Friedman, 1998). The crux of the analysis depends establishing that the total logeliiel of the non-missing nodes in the data set either increases or stays the same in each iteration of the algorithm. That is,
Σ logg(1 1 (0+H) >
(8) log (T. (1)))
(27)
To that end, note that
n
( Σ log pog 1 (010)) = Σ
log pa logp(z
(1 (1) (.. JR. (0))) Σ
log pl p( 12 (1) O).
Taking expectation with respect
Fr. ni
(0)
( on both side, we get
log p( T. (0))) = Σ
E (0)
oo e e logg(af), ( (0 ())
n.
10 |=t
(0) or log log((1/))
(1) (6)
( (0)
E
e
or
log
10 1(1, x(0,0). (28)
=Q(e)or)
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The first term the RHS in the above equation is nothing but Q(+)e'). This is maximized in the M-step, T.e., Q(01@(11) 2 Q(010'). On the other hand,
p r. r()) ,0t11) -DLL (p(= (0)
(8) (i)
PT, (')lp'()(( (1) ( 10, ,000, = $0.
(0) B d0 1 e
log
p 2 (1)
T, (C)
(8)
02
Thus,
Σ
E ( p log p( 11 nn
2T; Tr 10, 0 < Σ E 2 pi log p( (0 (1) 2T; (0) O³)
(29)
From combining eqs. (28) and (29), we can see that the end of the M-step eq. (27) satisfied. Similar the previous results EM convergence (Wu; 1983; Friedman, 1998), MissNODAG reaches stationary point of the optimization objective.
B.3 Consistency of MissNODAG
In this subsection establish the theoretical foundations for proving consistency of exact maximization of the MissNODAG score function. We begin by reviewing the relevant definition and prior results necessary to prove theorem 2.
B.3.1 Background
Consider a directed graph 9 EX, A path π between nodes X; and Xx is a sequence (Xigr Xix.... ........ X with X,, X4, Xi, Xg. and every two consecutive nodes in the sequence i.e., either Xi, + Xi511 E E or Xi, <- Xiy41 E E for all j O,..., - 1, with X. A path is directed if all the edges from X, to Xx are oriented the same way. A cycle through node X, consists of a directed path from X to a node X, and the directed edge X, + X,, For a node X;, the set of ancestors is defined as (X,, := (X, E X | a directed path exists between X, and X,). Similarly, the descendant of a node is defined as (X,, = (X, E X a directed path exists between X, and X2). The strongly connected component of node X; is given by the intersection of the ancestors and the descendants of X;, i.., scg aug(X,) n deg (X;). We can apply these definitions to subsets Xu by taking the union over all the elements in the subset, i.e., deg (Xu) (X..). Finally, node X; is called collider in a path n if it satisfics the following two conditions: (i) it is a non-endpoint node, and (ii) the subpath (X,, X,, X2) is of the form X + X X.
connected by an edge,
an acyelic graph, the notion of d-separation used to relate structural properties in the graph & to independencies in generated distribution. This known as the directed global Markov property (Forré Mooij, 2017). However, in general, cyelic graphs do not obey this property and generalization of d-separation in the form of o-separation was proposed by Forré & Mooij (2017) to characterize the relationship between graphical properties in cyclic graph to the independencies observed in the distribution.
Definition B.1 (o-separation). Let G = EX be directe graph and let Xc C X be subset of nodes. A path π = -X,,, Xi,,) is said to be o b-blocked given Xe if
1. the first node of x, Xio E Xx or ils last node Xi, E Xc,,
contains a collider Xj Xc,
π contains a non-collider X; E Xc that points towards a neighbor that is not in the same strongly connected component as X, in 9, i.e., X2 + X, E a and X & sog(X;), or X, + Xx E and X & 8eg(1 (X;).
The path π is said be o-open given Xe if is not bloockedd. Two subsets of nodes XA. S X said to be a-separated given Xe if all the paths between Xa and Xo, where a E A and b E B, are -blocked given Xc, and is denoted by
XA 1 X Xc-
g
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C2
X1
X2
X3
X1
X2
X3
X1
X2
XX
X2
X1
R1
R2
Ra
Rg
Ra
R1
1
1
Y1
Y2
Ya
Ya
Yi
Y3
9m
(G.))))(
(o(.)))(G..)
Figure B.1: Illustration of the augmented graph 9hh corresponding to the family of interventions (X2) The graphs (9m)))((m) and (9m)))(9m)) corresponding the mutilated graph obtained after hard interventions X2 and X3 respectively. The augmented graph 9 is the union of the graphs Im, (om))))Gmm, and (o(X)) with the addition of the context variables C₁ and C2.
o-separation defined above reduces to d-separation in acyelic graphs. We now define the generalized version of directed global Markov property for cyclic graphs.
Definition B.2 (General directed global Markov property (Forré & Mooij, 2017)). Let G = (X,E) be directed graph and denote the probability density of the observations X. The probability density p satisfies the general directed global Markov property if for XA. S
XA X Xc = XALXE Xc,
p
where XA -p XB Xc. denotes conditionally independence of XA and XB given X with respect to p.
Note that all the statements made above doesn't exactly depend on the vertex set. Thus, we can replace G with the m-graph 9m and retain all the properties stated carlier on the new graph with the vertex set (X, R).
B.3.2 Joint causal modeling under interventions
To unily multiple interventional settings, we adopt the joinl cousal model Mooij et al. (2020). We create a meta system by augmenting the system variables R) with exogenous context variables C = (Ci...., Cm), where each Cm corresponds
specific intervention target set Xz..- Cn @ for all m = 1,.... M denotes the observational setting. We construct augmented graph 9 where the children of each context variable are its targets, that is, cCm(Cm) Xz.. (see figure B.1). Given family of interventional targets and the corresponding context variables, the structural equations for X take the form:
(Cm)k. (e((agg(xi) + ex: otherwise.
if m E [M] s.t. X E Ims and Cm * 0,
fa(pagg (x)
Note that pagz (Xx) includes nodes from both X and CH. The context distributions defined over the context. variables CZ not influence the system any meaningful way, as noted by Mooij et al. (2020). Moreover, we assume access to the context distribution as the interventional settings are known apriori. For the meta system defined above, the observational distribution corresponds to Pg=(X Ci interventional distribution for the interventional targets XI,, corresponds to pGz (X C. Ex... C-m 0), i.e.,
C. 0). Similarly, the
Pgz (X Om = Er,, C-m = 0) (X..
Here, Pdu(I..)((G..) corresponds to the interventional distribution generated by Gm for the interventional setting NT.. Furthermore,
Pgr (O XX) xx== P (C )PT (X C*)).
(30)
Note that the missingness mechanism is unaltered by augmented the graph with context variables. Thus,
Pgz (X, R, OF) Pggz (C2 )oz (X c p).... X).
(31)
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Recall, for the interventional setting XI.. the probability density function governing the observations X is given by eq. (4), which we repeat here for convenience
Pdo( = pr(X))pp det (J)))
Definition B.3. Let G (X,E) be a directed graph, and {XIm im with XI, X be family of interventional largels. Let Mz(9m) denole the set of posilive densilies Pez. RRdd + R such that Pez. is given by eq. (31) for all F IRd + RR, with fi(X) = fi(pag(X (X;)), such that the resulting forward map (id - F) is invertible, for all (of....,of}) such that e e~ NO, Pog, (R XX= IR22 R.
and positive densily
Proposition B.4. Let G (X,8) be a directed graph, and {XIm Im= with XI, = X be a family of interventional targets, let p Mr))m), then p satisfies the general directed global Markov property relative to GTh
Proof. For directed graph 9 and a choice of F: Rd I such that the forward map (id FF) is invertible, the structural equations are uniquely solvable with respect to each strongly connected component of G. Moreover, the addition of context variables and the missingness indicators in the augmented graph does not introduce any new eycles. Therefore the meta system forms a simple SCM (Bongers et al., 2021). Thus, from Theorem A.21 in (Bongers et al., 2021), the distribution Poz, is unique and it satisfies the general directed global Markoe property.
We now define the notion of interventional Markov equivalence class for DMGs based on the set of distribution generated by the DMGs.
Definition B.5 (I-Markov Equivalence Class). Two directed graphs 9m and I'm are I-Markou equivalent if and only if Mz(9m) Mz(mmm) denoted as 9m =I Ptm- The set of all directed mixed graphs that are I-Markov equivalent to Gm is the I-Morkov equivalence class of Gm, denoted I-MECIGm).
B.3.3 Full law identification from observed law
From the chain rule of probability, the full law p(X, R) is identified if and only if the missingness mechnanism p(R) X) is identified. That is,
1) P(X, R) p(R 1 x p( x).
(32)
Nabi et al. (2020) showed that when the m-graph satisfies assumption 1, the missingness mechanism is indeed identifiable, with the identifying functional given by
K II k=2
1
p(R X
P 1=1
R X)
ORR Rec RSA 1,,,,
(33)
where Z the normalization constant, R_ R RR. Rek (RI..... Rk-11, Ryb {RR41.....
Rx and
OR(Re, Rek RS* = 1, p( Re Rsb 1, X)
x p(R =11 1| = 1,X) pRRs = RS* = R, p(R, 1,
Note that, under assumption 1, all the terms the rinht-hand side the equation above are functions of the observed distribution, and thus can be estimated from the available data. Furthermore, we make the standard assumption that the missingness mechanism is strictly positive. In the following proposition, we establish that the map from observed law to the full law is continuous and injective.
Proposition B.6. Let Gm. be a directed graph satisfying assumption 1. Then, the map V Pobs + Pfolt given by eq. (32) is injective and continuous.
Proof. Injectivity the map & a direct implication of that fact that the missingness mechanism in eq. (33) is identifiable given the observed law, see (Nabi et al., 2020) for more details. Furthermore, continuity of yp can be concluded from the fact that the identifying functional in eq. (33) is made up of multiplication and division of positive densities.
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B.3.4 Proof of Theorem 2
We now provide the proof of the main theoretical result of this paper. Recall the score function introduced section 3.1,
S(gm)
sup Σ E (Xr (Xr. (m) Poo R) >1gml
((m) where, Pobs is the observed part of the data-generating distribution for XIm E {XImm 1M m= and Og... (0gm PGm.) represents the model parameters. In the context of the meta system, the score function above is equivalent to the following score:
S2(9m) == sup E logPog (Xr, R, C @g... A/9ml: Bp.. (Xr (X),
(34)
where, Pgz, (xr,R, (...) corresponds to the restriction of the complete density function, given by eq. (31), to only the observed nodes. The density pł denotes the joint ground-truth distribution for the observed and the context variables. We define (9( be the set of all distributions Pez (X, C () that can be expressed by the model specified by eqs. (3), (13) and (18). That is,
(9) (p) 3 s.t p = Poz, (-10))
(35)
Note that, Pr(9m) Mz(gm)) Theorem 2 relies on the following set of assumptions. The first one ensures that the model is capable of representing the ground truth distribution.
Assumption B.7 (Sufficient Capacity). The joint ground truth distribution pz is such that pz E Pr(M)), where 9 is the ground truth m-graph.
other words, there exists a θ such that p1 = P (-1 The second assumption is a generalization of the standard faithfulness the interventional setting for cyclic graphs.
Assumption B.8 (I-o-faithfulness). Let V RX ),), for any subset of nodes A,B.DCVUC
and Im E I
A G
B
B -
VA L VB Vo. Poz
As a result the assumption above, any conditional independency observed in the data must be a consequence of a a-separation in the correspouding interventional ground truth graph. Assumption B.9 (Finite differential entropy). For a family of interventional targets {Xr.. JM m==
ppg IX, co.
The above assumption ensures that the scenario where S(G*) and S(G) are both infinity is avoided. This is formalized in the lemma below taken from (Brouillard et al., 2020). Lemma B.10 (Finiteness of the score function (Brouillard et al., 2020)). Under assumptions B.7 and B.9, [Sz(g) < co.
We finally make the assumption that the density strictly positive. Assumption B.11 (Strict positivity). For all V = (X, C, R), Paz (V) > 0 for all Poz E MM(99m).
For the form of score function in eq. (34), Brouillard et al. (2020) showed that the difference score function between 9t and 9m can be expressod as the minimization of KL diverengence plus the difference in the regularization terms.
Lemma B.12 (Rewritting the score function (Brouillard et al., 2020)). Under assumptions B.7 and B.9, have
S(g - S(Gm. = inf (p ( ()) + A(19ml - 19
e
where (-)) denotes the distribution restricted to only the observed variables.
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Furthermore, under complete data, Sethuraman & Fekri (2025) showed that the KL divergence appearing the in lemma above is strictly positive.
Lemma B.13 (Lemma A.18, (Sethuraman & Fekri, 2025)). Let G (V,E) be directed graph, for set of interventional targets {XIm. J and p' MM(Gmm), then
inf
D(p" lp) 0.
peMr((m))
We now extend this result to the case of missing data. To that end, denote MI,obs to be the set of observed densities generated by the graph Gm.
Lemma B.14. Let G (V,8) be directed graph, for a set of interventional largels {XT,m, }Md m= and Pobe & obe (mm),, then
inf Dob EotoEM (Gm)))
D(PoseelPosa 0.
Proof. From lemma B.13, we know that p" & Mz((9m) then for the full distribution we have
inf peMz(Sm.))
D(p (p)
This implics that p" does not conform the independencies enforced by the a-separation constraints of 9m- Let us now assume, for the sake of contradiction, that the infiruum of the score on the observed data is zero:
inf D(PosellPar)) = Pobs ob.
By Pinsker's 's incquality (Cover, 1999), this implies convergence total variation and thus convergence in distribution. Therefore, there exists sequence of distributions ) Mzzo (9m) such that:
(k) lim Pobs Pohs
Let V be the identification mapping in eq. (32) such that (P = p. From proposition B.6, V is injective and continuous. By applying the continuous map the sequence we get:
lim (P((ot) 10000
lim Poos) -
(p =
Where the last equality can be obtained by invoking injectivity of the map W. Let g(d) = (+)). Since (k) Pobe is generated by the model Gm, its identified full law (()) must be a valid distribution in the full model Mr(9m).
We now have sequence ()) c M(Gm) that converges to p². This would mean that
inf D(p* 0.
pe (eM)((
which is a contradiction. Thus, have proved the lemma.
We are now ready prove Theorem Recall,
Theorem 2. Let {XT.m MM be a family of interventional largels, let 9;, denole the ground truth directed mixed graph, p(k) denote the data generating distribution for Ig, and gm arg maxg Egu Egu S(Gm). Then, under the Assumptions 1, B.7, B.8, B.9, and B.11, and for suitably chosen 0, we have that Gm = G That is, 9m is Mararoov equivalent to G
Proof. The proof follows that of Theorem 2 Sethuraman & Fekri (2025), with lemma B.14 replacing lemma B.13; reproduce it here for completeness.
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It is sufficient to show that for 9m & I-MEC(GM). the score function of 9m is strietly lower than the score function of Im- i.e., S(Fm) > ((Gm). Since 9m & I-MEC((M) and pł E MMI(9 (by assumption B.7), there must exist subsets of nodes A, D U URUCE CD such that either:
A
B and A L ALBID, 9m
(C1)
9.
or
A / B and A B D. Gm
(C2)
9m
If no such subsets exist, then 9m and Fn impose the same a-separation constraints and thus induce the same set of distributions. This would imply that 9m E I-MEC((G)). contradicting our assumption. Since P E MMz(9m), in the case of (C1), it must be true that A Kp; B D (assumption B.8). Therefore pž doesn't satisfy the general directed Markov property with respect to G and hence Př MM(Gmm) For (C2), if p Mm)) then from assumption B.8, it must be true that A Lpz B D. However, PT MMr())) and proposition B.4 implies that A Lp; B D, resulting in contradiction. Therefore, pł & &Mz(9m). Moreover, from proposition B.6 we can conclude that P7 & MMI,obsGGni
For convenience, let
m(Gm) == inf DEL (oz (DT Poo. (: 18)).
Note that
(( inf DKi. (ar, logz, obol |0) inf
D PKI,P2,, (Ip) > 0,
oe (G.)
where we use lemma B.14 for the final inequality. Thus, from lemma B.12
S(Gg)) - S(Gm) m((m) A(1sml - 19
(36)
Following (Brouillard et al., 2020), we now show that by choosing À sufficiently small, the above equation is
stictly positive. Note that |9ml 9 then
O. Let G+ (9m 19ml < 19 Choosing
A such that 0 < A < ming c G...
(Gm) we see that:
A min gmEG+
(37)
9
[mll
(9m)
VGm G
(38)
9
19ml
()ml)) < m1((m))) Vg. E G+
(39)
0 < m)(m) A(19ml - (9 S(9h)) S(9m) 4gm E C+.
(40)
Thus, every graph outside of the general directed Markov equivalence class of (G )z has strictly lower score.
C Additional Experiments
C.1
Sensitivity to Model Parameters
We conducted an scnsitivity analysis comparing MissNODAG with bascline models to assess scnsitivity to: (a) number of samples per intervention in the training set, and (b) average edge densily of the target law graph. In all cases, graphs were generated using the ER random graph model. Training data consisted of single-node interventions over all nodes in the graph. Exogenous noise variables and m-graphs were sampled as described in section 4. In both the cases we generate the data from the nonlinear SEM (B 1). Results (averaged over 10 trials) are shown in figure C.1.
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Varying Interventional Training Samples
Targel Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
20
SHD
10
SEH
500
1,030
1,500
2,000
2,500
500
1,000
1,500
2,000
2,500
500
1,000
1,500
2,000
2,500
Samples/Tune /Intevventionnnn
Samples/Ineerventhn
Samples/Intervente
Varying Target-Law Edge Density
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
20
SHD
SEID
10
5
Edge density
Edge density
Edge density
Mis:NODAG
MisDAAG
Missforest
Op-Transport
Mean
Logistic
Figure C.1: Performance comparison between MissNODAG and the basclines as function of: (top row) number samples per intervent ion in the training data, and (bottom row) edge density of the target-law graph.
Varying interventional training samples. With the target law and missingness mechanism outgoing edge densitics fixed at 2 and 3, respectively, we varied the number of training samples per interventional setting from 500 to 2,500 (in increments of 500). The average missingness probability was set to 0.3. As seen from figure C.1 (top row), MissNODAG outperforms the baselines in recovering both the target law and the missingness mechanism. NissNODAG achieves near perfect larget-law recovery even with 500 samples per intervention. This is showcases the superior sample requirements of MissNODAG compared the basclines.
Varying target law edge density. In our experiments, the outgoing edge density of the target law varied between 1 and 4, while the average missing probability was fixed at 0.3. Each interventional setting comprised 1,000 samples. From figure C.1 (bottom row), we observe a trend consistent with the previous experiment with MissNODAG outperforming the baselines. However, we observed that the models performed comparably terms of target law and + R edge recovery.
C.2 Misspecification
We consider three kinds of model misspecification: (i) soft interventions, (ii) non-identifiable missingness mechanism, and (iii) mismatched parametric family for missingness mechanism. In each selting, we considered nonlinear SEM with K 10 nodes, missing probability to 0.3 and provided all single node interventions as training input.
Soft interventions.
In this setting, the interventional structural equation model is set as follows:
X₂ = (x(op((X))) + if Xx Xx
(e(ppg(Xx)) + Xx $ Xx.
our case, we set fx = fx and consider e ~ NO00 of), é Ni,,, of), with μ4 1 (also known as shift interventions). Thus, the parent-child interactions are preserved under intervention. The results are summarized in figure C.2. As seen from the figure, MissNODAG achieves comparable performance to that of the baselines, albeit achieving higher SHD score compared to hard interventions.
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Target
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
Missforest
Method
Op- Transport
Mean
O
ENCO
MVPC
Logistic
10
20
5.0
7.5
10.0
SHD
SHD
SHD
Figure C.2: Performance comparison under soft interventions
Non-identifiable missingness mechanism. Here, we explicitly add self-censoring edges as well colluders into the missingness mechanism while considering hard interventions. The results are summarized in figure C.3. As seen from the figure, MissNODAG is still capable of learning the target law, however, the absolute SHD score increases for missingness mechanism recovery.
Target Law
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
MissNODAG
10
Missforeel
Method
Op-Tramsoort
Mean
CIH
ENCO
MVPC
Logistic
10
20
30
10
15
20
2.5
5.0
7.5
10.0
12.5
SHD
SHD
SHD
Figure C.3: Performance comparison on non-identifiable missingness mechanism
Mismatched parametric family for missingness mechanism. In this setting, we consider the condi- tional distribution corresponding each missingness mechanism R2 be follows:
p( R4 pag... (Rx), Φn) expit (N, R) + zx).
where (X,, It) is a 2-layer neural network with tanh activation. The results are summarized in figure C.4. As seen from the, the absolute SHD score for target law recovery has increased compared to the well-specified case, however, MissNODAG still retains a comparable performance to that of the baselines.
Target
Missingness Mechanism (X2R)
Missingness Mechanism (R2R)
NODAO
Missforest
Method
Op- Transport
-+
Mean
+DO
ENCO
MVPC
Logistic
20
30
15
20
SIID
SIID
SHID
Figure C.4: Performance comparison under missingness mechanism misspecification
C.3
Computation Time and Convergence Analysis
In figure C.5, we compare the training times of MissNODAG and the baselines for two settings: 10 nodes (left) and K 20 nodes (right). In both cases, the training data consists of single-node interventions all
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K = 10 nodes
K 20 nodes
Time (sec)
200
Time (sec)
600
400
100
200
MissNODAG
MissDAG
op-Transport
Missforest
Mean
NODAGS-cclen
Figure C.5: Training time comparison
nodes, with 500 samples intervention. MisSNODAG and MissDAG were trained for 100 epochs. For other baselines, the reported time includes both data imputation and subsequent training of NODAGS-Flow on the imputed data for 100 epochs. We also report the training time of NODAGS-Flow itself. The two primary computational bottlenecks in MissNODAG are the rejection sampling procedure and the computation of the Jacobian dog-etterm,,ant, which is required during both the imputation and target law update stages. Utilizing power series approximation yields complexity of O(K2) for the log-determinant, which can be further reduced in practice via the Hutchinson trace estimator. Consequently, the overall time complexity of MissNODAG is L), where L denotes the maximum number of rejection sampling steps.
As shown in figure C.5, MissNODAG requires the longest training time, though only slightly more than MissDAG in both settings. This additional time is primarily due to jointly learning the missingness mechanism, which MissDAG does not model. As expected, compute time approximately doubles when the number of nodes is doubled.
Additionally, report the training and eval metrics (log probability of observations, colluder constraint function value, DAG constraint on R + R edges, and target law SHD) as function training ilerations in figure C.6.
Log Probability
Constraints
0.15
Ao (a)
5
0.1
log p(X)
5 10
15
log p(12 X))
300
600
900
1,200
1,500
300
600
1,200
1,500
Iterations
Iterations
Acceptance Rate
Target Law (SHD)
0.53
20
15
10
0.51
0.5
300
600
900
1,200
1,500
600
1,200
1,500
Iterations
Iterations
Figure C.6: Training and evaluation metrics as a function of training itcrations. Top left: log probability of the observed data for both target law and missingness mechanism. Top right: DAG constraint for R R edges, and colluder constraint for missingness mechanism. Bottom left: acceptace rate per iteration for rejection sampling. Bottom right: Target law SHD.
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C.4 Sachs Protein Signaling Dataset
We evaluate MissNODAG biological dataset for protein signaling network discovery Sachs et al. (2005), widely used benchmark for causal discovery algorithms.
The dataset contains continuous measurements of multiple phos- phorylated proteins and phospholipid components in human immune system cells, with the corresponding network capturing the ordering of interactions among pathway components. Basod 7466 samples across (2005) identified 20 edges in the underlying graph. Using the consensus network from Sachs et al. (2005) as ground truth, we evaluate performance using the Structural Hamming Distance (SHD) as the error metric. Since the data doesn't contain any missing values, we define a synthetic missingness mechanism generate missing data from the complete dataset. We the missing probability to 0.3. The results summarized in table C.1. We note that the ground-truth graph from (Sachs et al., 2005) is itself DAG, despite feodback loops being known lo exist in these systems (Sturm et al., 2011), as result, out SHD scores are conservalive with respect to MissNODAG's true recovery performance. The recovered directed graph is visualized in figure C.7.
Table C.I: Performance comparison on
Sachs et al. (2005) protein signaling dataset.
11 cell lypes, Sachs et. al.
Method
TL-SHD
MisNODAG
20
MissDAG
23
Optransport NODAGS
22
Optransport ENCO
23
True Graph (TL)
MisNNODA: Estimated Graph (TL)
Ref
Ref
PLCk
PLCx
PIP2
PIP2
PIP3
PIP3
Erk
Erk
Akt
Akt
PKA
PKA
PKC
PKC
p88
p38
JNK
JNK
Mak
PLOC
PIP2
PIP3
Eec
Ala
PKA
PKC
p3s
JNK
Ref
Mek
PLCC
PIP2
pIF3
Brr
Aks
PKA
PKC
1988
INN
Figure C.7: (Left) Consensus graph from Sachs el al. (2005), (right) graph learned by MisSNODAG.
D Implementation Details
In this section, we describe the implementation details of the MissNODAG framework and the baseline models used for performance comparisons.
MissNODAG. We implemented our framework using the Pytorch library in Python and the code used in running the experiments can be found in the codes folder within the supplementary materials. We plan to make the code publicly available on GitHub upon publication of the paper.
Starting with an initialization of the model parameters O we alternate between the E-step and M-step until the parameters converge. In the E-step, algorithm used for imputing the missing data, followed by maximizing the expected likelihood of the non-missing nodes in the M-step. We follow the same setup as Sethuraman al. (2023) for modeling the causal functions, i.c., neural networks (NN) along with dependency mask with entries parameterized by Gumbel-softmax distríbution, and for computing the log-determinant of
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the Jacobian, i.c., power series expansion followed by Hutchinson trace estimator. Poisson distribution is used for ps for sampling the number of terms in the expansion to reduce the bias introduced while limiting the number of terms in the power series expansion of log-determinant of the Jacobian, see section 3.3. The final objective in the M-step is maximized using Adam optimizer (Kingma & Ba, 2014).
The learning rate in all our experiments was set to 10-2, The neural network models used our experiments contained one multi-layer perceptron layer. No nonlinearities were added the neural networks for the linear SEM experiments. We used tanh activation for the nonlinear SEM experiments and ReLU activation for the experiments the Bertur-CCITE-see data set. The regularization constant À was set to 10 for the synthetic experiments and 10-3 for the Corturb-CTTEsse experiments. All experiments were performed on NVIDIA RTX6000 GPUs.
Baselines, For the baseline NODAGS-Flow' we modify the code base provided by Sethuraman et al. (2023) use the imputed samples for maximizing the likelihood. The hyperparamcters of NODAGS-FloW was set to the values described in the previous subsection.
Missforest imputation is performed using the publicly available python library missingpy'. We use the codebase provided by Muzellec et al. (2020) for optimal transport imputation", and the codebase provided by the authors was used for MissDAG' (Gao et al., 2022). ENCO (Lippe et al., 2022) was implemented using the codebase provided by the authors with the default hyperparameters. Finally, MVPC was implemented using the causal-learn' python library (Zheng et al., 2024). The default parameters are used for Missforest, optimal transport imputation, and MissDAG.
hntpee///gttabb.co//palppppppppp
httpe://githau..oo///sst///////s
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Reviews:
[{"review_id": "JVZhtBjd9p", "reviewer": "Reviewer_hP9A", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes MissNODAG, a differentiable framework for learning cyclic causal graphs from incomplete interventional data under MNAR mechanisms. The work extends EM-based causal discovery to simultaneously handle both cyclic structures and graphically-represented MNAR mechanisms, building upon NODAGS-Flow and MissDAG. The technical approach employs residual normalizing flows with rejection sampling for imputing missing values in cyclic SEMs, and uses augmented Lagrangian methods to enforce identifiable MNAR constraints including no self-censoring and no colluders. The authors provide convergence analysis showing the algorithm reaches a stationary point, and demonstrate superior performance over imputation-then-discovery baselines on synthetic data and gene perturbation datasets.The paper addresses an important problem since real-world systems often contain both feedback loops and missing data, making this a timely and relevant contribution to the causal discovery literature.", "claims_and_evidence_yn": "No", "claims_explanation": "While the theoretical framework is sound and the presentation is generally clear, several critical gaps undermine the strength of the evidence.\n\nQ1. The identifiability claims in Section 3.1. The paper states that with complete data, Sethuraman & Fekri (2025) showed the recovered graph belongs to the correct interventional Markov equivalence class, and then suggests similar performance holds under missingness. However, this represents a logical gap. The EM algorithm does not directly optimize the observed data likelihood but rather optimizes a lower bound, which fundamentally changes the optimization landscape. The authors need to either prove that EM stationary points correspond to the correct equivalence class under their missingness assumptions, or provide extensive empirical evidence demonstrating when and why this gap is negligible. Without such analysis, the central claim that MissNODAG recovers the correct causal structure remains inadequately supported.\n\nQ2. The synthetic experiments use only K=10 nodes, which is relatively small and insufficient to demonstrate scalability. A more systematic evaluation over larger graph sizes—analogous to the node-scaling experiments in Section F.5 of MissDAG—would be necessary to assess how performance degrades as problem complexity increases. The ablation study in Appendix C.1 tests up to K=30 and shows performance degradation, but this is not analyzed or discussed. Furthermore, the paper includes only a single real-world dataset from gene perturbation studies, leaving open questions about generalization to other domains where cyclic relationships and MNAR missingness are relevant.\n\nQ3. The baseline comparisons raise concerns about fairness and completeness. In particular, comparing against MissDAG on cyclic graphs is inherently problematic, as MissDAG is explicitly restricted to DAGs. In addition, the experimental evaluation does not include methods specifically designed for MNAR settings. More recent deep learning–based MNAR imputation approaches, followed by graph learning on the imputed data (i.e., impute-then-learn pipelines), are absent.", "audience_yn": "Yes", "audience_explanation": "Missing data is important in domains such as biology, healthcare, and economics, where cyclic relationships (e.g., gene regulatory feedback or economic loops) are also common. Most existing methods handle either missingness or cyclicity, but not both. The proposed approach directly addresses this practical gap.", "requested_changes": "C1. Identifiability Analysis (Section 3.1)\n\nThe identifiability claims under missing data require further clarification and support. The authors should either\n- formally establish that stationary points of the EM objective correspond to the correct interventional Markov equivalence class under the assumed missingness mechanism, or\n- provide systematic empirical evidence demonstrating when and why the gap between optimizing a variational lower bound and the observed-data likelihood is negligible in practice.\n\nIn addition, the assumptions required for identifiability in the presence of missing data should be stated explicitly and discussed more clearly.\n\nC2. Experiments (Section 4)\n\n- Scalability: Add experiments with larger graphs (e.g., (K=20, 30, 50)) and report computational cost (runtime and memory) as a function of (K). Any observed scalability limitations should be discussed transparently.\n\n- Real-world evaluation: Include at least one additional real-world dataset.\n\n- Baseline comparisons: Improve fairness and coverage of baselines by including MNAR-specific methods or other recent approaches designed for MNAR/cyclic settings. It would also be informative to compare against (a) methods that handle cyclic graphs with complete data, and (b) methods that handle MNAR missingness but assume DAGs. The paper should explicitly acknowledge that comparing MissDAG on cyclic graphs is inherently unfavorable to MissDAG.\n\n\nC3. Logistic MNAR Assumption \n\nThe choice of modeling (p(R \\mid X)) using logistic regression, an ablation comparing logistic regression against a small MLP (e.g., a 2-layer network) for (p(R \\mid X)) would strengthen the empirical support of the robustness to model misspecification, particularly when the true missingness mechanism deviates from a logistic form.", "broader_impact_concerns": "I do not see any significant broader impact concerns. The proposed method focuses on methodological advances in causal discovery under missing data and does not raise immediate ethical, safety, or societal risks beyond those common to data-driven modeling approaches.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "cQwQ13Si7l", "reviewer": "Reviewer_FejH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper aims to tackle two important problems in causal discovery: (1) handling missing values, and (2) extending the setting to cyclic graphs. To do this, the authors build on MissDAG and use an iterative procedure that interleaves imputation and likelihood maximization to learn both the (possibly cyclic) structure and the missingness mechanism. The evaluation is conducted on synthetic data and real-world single-cell RNA-sequencing data. Overall, I think the paper is technically solid, though some theoretical aspects and parts of the evaluation would benefit from further clarification.\n\nThe key strengths are that the paper addresses a genuinely important problem—both missing data and cyclic structures frequently arise in real-world problems and the technical development looks sound; I went through the main components in detail and the methodology seems well-constructed.\n\nAs for weaknesses, I would think of two main things: the connections to prior work could be discussed more clearly, and the theoretical identifiability conditions remain a bit vague. Additionally, the evaluation section could be improved (right now is only evaluating on prediction).", "claims_and_evidence_yn": "Yes", "claims_explanation": "**Strengths**\n\n- I think that although the overall framework is largely built from common modules in causal discovery, such as the used objectives in EM and imputation, as well as contractive residual flows that are widely used for nonlinear additive models, combining these pieces into a single method that works for this specific and realistic setting (cyclic graphs with MNAR missingness) is still non-trivial. I didn’t find any critical issues in how these components are put together. \n\n- The problem setup itself is well motivated, and the synthetic results are convincing to me.\n\n**Weaknesses**\n\n- The most significant one is that the identifiability argument feels unclear or somewhat informal at the moment. I realize this may rely on Sethuraman & Fekri (2025), but I think the paper would benefit from adding a more formal proof or a clearer statement of the identifiability conditions. \n\n- Relatedly, it would be helpful to clarify whether this work (especially the MNAR component)can be viewed as a special case of Sethuraman & Fekri (2025), or how exactly it differs.\n\n- On the real-world evaluation, I understand that there is no ground-truth graph. Still, since this is causal discovery (not just prediction), it would be important to show and discuss the learned structure and check whether at least part of it aligns with existing domain knowledge. I think this would make the evidence that the method is (partially at least) recovering meaningful causal relations.\n\n- A relevant missing baseline  `Lippe, Phillip, Taco Cohen, and Efstratios Gavves, “Efficient neural causal discovery without acyclicity constraints,” ICLR 2022`.", "audience_yn": "Yes", "audience_explanation": "I think both the causal discovery part and the potential application on RNA sequence data will be of interest to the community.", "requested_changes": "- A formal identifiability proof \n\n- A clarification on the difference between Sethuraman & Fekri (2025)\n\n- For real-world evaluation, show the learned graph structure and analyze whether it aligns with the domain knowledge \n\n- More baseline comparison as mentioned (though the one I mentioned is not for missing data, you can combine it with imputaion)", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Oqz8AXRnDT", "reviewer": "Reviewer_DQNF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a unified framework for learning causal graphs with cycles jointly with an MNAR missingness mechanism using a differentiable EM algorithm. It combines contractive flows for the substantive model with a constrained DAG over missingness indicators, achieving identifiability under specific assumptions. The empirical results show improvements over existing approaches on both synthetic and a biological dataset.\n\n**Strengths:** real-world motivated problem, clear formulation, principled identifiability assumptions, well-designed EM procedure, strong synthetic performance.\n\n**Weaknesses:** strong assumptions on interventions and missingness structure, limited real-world stress-testing of MNAR assumptions, and scaling concerns for the E-step.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The theoretical assumptions are clearly stated, the EM derivation is sound, and synthetic experiments validate the core claims. The real-world dataset analysis is less conclusive due to simplifying assumptions about missingness but still informative.", "audience_yn": "Yes", "audience_explanation": "Causal discovery with MNAR missingness and cycles is a challenging and active research area; the paper addresses a gap in current methods and provides a principled solution.", "requested_changes": "Rather critical:\n- Clarify rejection-sampling details and provide diagnostics (acceptance rates, constraint activity).\n- Provide experiments with partial or weak interventions to test robustness.\n- Add discussion and/or experiments evaluating violations of the “no self-censoring” and “no colluders” assumptions.\n- Discuss practical limitations more explicitly.\n\n\nNon-critical but valuable:\n- Report convergence behavior and computational cost.\n- Broaden evaluation metrics (e.g., causal effect recovery).\n- Release anonymized code for reproducibility (would be valuable for reviewing process)\n- Add sensitivity analysis for treating zeros as missing in the Perturb-CITE-seq dataset.", "broader_impact_concerns": "The method relies on interventional data and strong assumptions on missingness mechanisms; misuse or misinterpretation may lead to incorrect causal conclusions if these assumptions do not hold in real applications (e.g., biology or healthcare). These concerns should be stated more explicitly in the Broader Impact section.", "additional_comments": "Overall, this is a strong and well-presented paper with meaningful contributions. Addressing the above points, especially around assumptions and robustness, would make the work more convincing and broadly applicable.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)