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arxiv: 2410.18918 · v2 · submitted 2024-10-24 · 📊 stat.ML · cs.LG

MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data

Muralikrishnna G. Sethuraman , Razieh Nabi , Faramarz Fekri This is my paper

Pith reviewed 2026-05-23 19:12 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords causal discoverycyclic graphsmissing datamissing not at randomexpectation maximizationadditive noise modeldifferentiable learninggene networks
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The pith

MissNODAG recovers both cyclic causal graphs and the missingness mechanism from partially observed data by alternating imputation with likelihood maximization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a framework that combines an additive noise model with an expectation-maximization loop to jointly infer cyclic causal structures and the process that causes observations to be absent, including cases where absence depends on the unobserved values themselves. Standard causal discovery methods assume either acyclic graphs or complete data, so this approach targets settings like biological networks where feedback loops and incomplete records are common. If the procedure succeeds, it produces consistent estimates of both the graph and the missingness parameters when the score is maximized exactly in large samples. The framework is implemented as a differentiable model that alternates between filling in missing entries and optimizing the observed-data likelihood.

Core claim

MissNODAG integrates an additive noise model with an expectation-maximization procedure that alternates between imputing missing values and optimizing the observed data likelihood, thereby recovering both the underlying cyclic causal graph and the missingness mechanism from partially observed data, including data missing not at random, and establishes consistency guarantees under exact maximization of the score function in the large-sample limit.

What carries the argument

The alternating imputation and likelihood-optimization loop inside a differentiable additive-noise-model framework that jointly updates graph parameters and missingness parameters.

If this is right

  • Causal graphs containing feedback loops become identifiable from incomplete records.
  • Missingness mechanisms that depend on the unobserved values themselves can be recovered alongside the graph.
  • Consistency of the recovered graph and missingness parameters holds when the score function is maximized exactly as the number of samples grows.
  • The same procedure applies to both synthetic data generated from known cyclic models and real gene-perturbation measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The differentiability of the framework opens the possibility of scaling the method to graphs with hundreds of nodes by replacing the inner optimization with gradient steps.
  • If the additive noise assumption is relaxed to other identifiable noise models, the same alternating structure might extend to non-Gaussian or heteroscedastic settings without changing the outer EM loop.
  • Success on gene data suggests the method could be tested on other domains where both cycles and non-random missingness appear, such as longitudinal health records or sensor networks.

Load-bearing premise

The observed data are generated by an additive noise model whose parameters and missingness mechanism can be recovered together by alternating imputation steps with direct maximization of the observed likelihood.

What would settle it

A large-sample simulation in which the true cyclic graph and missingness parameters are known but the alternating procedure returns inconsistent estimates even when the score is maximized exactly at each iteration.

Figures

Figures reproduced from arXiv: 2410.18918 by Faramarz Fekri, Muralikrishnna G. Sethuraman, Razieh Nabi.

Figure 1
Figure 1. Figure 1: Example m-graphs with three variables illus￾trating: (a) An MNAR mechanism considered in our MissNODAG framework; (b) An MNAR mechanism where Rs are connected and the full law is identifiable. these graphs by Gm(V ), where V = (X, R, Y ). Two examples of missing data graphs (or m-graphs), with K = 3 substantive variables, are provided in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of results for learning causal graph structure (target law) under linear (left) and nonlinear [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of results for learning causal [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of target law recovery for linear SEM with varying training set sizes. The average missing probability was set to 0.2, and each Rk has a parent set cardinality of 3. 5000 10000 15000 20000 25000 # samples 1.5 2.0 2.5 3.0 SHD Nonlinear SEM (ER-1) 5000 10000 15000 20000 25000 # samples 5.0 5.5 6.0 6.5 Nonlinear SEM (ER-2) nodags+clean missnodag missforest optransport [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 6
Figure 6. Figure 6: Results of target law recovery for nonlinear SEM with varying training set sizes. The average missing probability was set to 0.2, and each Rk has a parent set cardinality of 3. D.2 Target Law Recovery: Performance as a Function of Cardinalities for paGm (Rk) We also evaluated target law recovery performance as a function of the parent set cardinality of the missingness indicators, which reflects the sparsi… view at source ↗
Figure 7
Figure 7. Figure 7: Results of target law recovery in linear SEM as the parent set cardinality of each Rk is varied. 0.1 0.2 0.3 0.4 0.5 Av. Missing Prob 2 4 6 8 SHD Nonlinear SEM (ER-1) 0.1 0.2 0.3 0.4 0.5 Av. Missing Prob 6 8 10 12 14 Nonlinear SEM (ER-2) |pa m (Rk)| 3 |pa m (Rk)| 4 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of target law recovery in nonlinear SEM as the parent set cardinality of each Rk is varied. D.3 Target Law Recovery: Learning DAGs from Partially Observed Observational Data Figures 9 and 10 present the results of learning DAGs from partially observed observational data. We followed the same procedure described in section 4 to generate the data, with the additional constraint that the re￾sulting gr… view at source ↗
Figure 9
Figure 9. Figure 9: Results of target law recovery for linear SEM when the target factorizes according to a DAG, with MNAR mechanism where Rk has a parent set cardinality of 3 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of target law recovery for nonlinear SEM when the target factorizes according to a DAG, with MNAR mechanism where Rk has a parent set cardinality of 3. D.4 Data Application: Gene Perturbation Here we present an experiment focused on learning causal graph structure corresponding to a gene regulator network from a gene expression data with genetic interventions. In particular, we focus on the Pertur… view at source ↗
Figure 11
Figure 11. Figure 11: Predictive performance over unseen interventions on Perturb-CITE-seq Frangieh et al. (2021) data. [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes MissNODAG, a differentiable framework combining an additive noise model with an EM-style procedure (alternating imputation and likelihood optimization) to jointly recover cyclic causal graphs and missingness mechanisms (including MNAR) from incomplete data. It claims consistency guarantees under exact maximization of the observed-data score in the large-sample limit and reports effectiveness on synthetic experiments plus a real-world gene perturbation application.

Significance. If the consistency result and the optimization procedure can be aligned, the work would address an important gap in causal discovery for cyclic systems with incomplete observations. The integration of differentiability with cyclic ANMs and MNAR handling is a potentially useful technical contribution, though its practical impact depends on closing the gap between the exact-maximizer theorem and the implemented alternating algorithm.

major comments (2)
  1. [Abstract] Abstract: The consistency theorem is stated only for exact global maximization of the score function. The described algorithm instead alternates imputation with gradient-based maximization of a differentiable surrogate over graph parameters and missingness mechanism. For non-convex observed-data likelihoods arising from cyclic ANMs, this procedure has no guarantee of reaching the global maximizer, so the theorem does not directly apply to the output of MissNODAG. This gap is load-bearing for the central claim that the method 'uncovers' the true graph and missingness mechanism.
  2. [Abstract] Abstract (and § on method): The paper does not appear to provide a proof or argument that the alternating optimization converges to the exact maximizer (or to a point whose implied graph is consistent) for the non-convex cyclic case with MNAR parameters. Without such a result or additional assumptions that rule out spurious stationary points, the consistency guarantee remains disconnected from the implemented procedure.
minor comments (1)
  1. [Abstract] The abstract refers to 'synthetic experiments' and 'real-world gene perturbation data' but provides no quantitative metrics, baseline comparisons, or controls for the missingness mechanism; these details should be expanded for reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for highlighting the important distinction between the consistency result under exact maximization and the practical alternating optimization procedure. We address the two major comments below and will revise the manuscript to clarify the scope of the theoretical claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The consistency theorem is stated only for exact global maximization of the score function. The described algorithm instead alternates imputation with gradient-based maximization of a differentiable surrogate over graph parameters and missingness mechanism. For non-convex observed-data likelihoods arising from cyclic ANMs, this procedure has no guarantee of reaching the global maximizer, so the theorem does not directly apply to the output of MissNODAG. This gap is load-bearing for the central claim that the method 'uncovers' the true graph and missingness mechanism.

    Authors: We agree that the consistency theorem applies strictly to exact global maximization of the observed-data score, while the implemented MissNODAG algorithm performs alternating imputation and gradient-based optimization of a surrogate, which offers no global optimality guarantee in the non-convex setting induced by cyclic ANMs and MNAR parameters. This is a substantive gap. In the revision we will modify the abstract, introduction, and theoretical section to state explicitly that consistency holds under the assumption of exact maximization (as currently written), and we will add a dedicated paragraph in the method section discussing the distinction, the non-convexity challenges, and the fact that the algorithm is a practical heuristic whose output may correspond to local optima. revision: yes

  2. Referee: [Abstract] Abstract (and § on method): The paper does not appear to provide a proof or argument that the alternating optimization converges to the exact maximizer (or to a point whose implied graph is consistent) for the non-convex cyclic case with MNAR parameters. Without such a result or additional assumptions that rule out spurious stationary points, the consistency guarantee remains disconnected from the implemented procedure.

    Authors: We confirm that the manuscript contains no convergence argument showing that the alternating procedure reaches the global maximizer or a consistent graph estimator in the non-convex cyclic MNAR setting. Deriving such a guarantee would require additional assumptions or analysis that are not present. In revision we will therefore weaken the language in the abstract and method description to avoid implying that the implemented algorithm inherits the consistency result, and we will include an explicit caveat about possible local optima and sensitivity to initialization, supported by the existing synthetic experiments that demonstrate practical performance. revision: yes

standing simulated objections not resolved
  • A proof or argument establishing convergence of the alternating optimization to the exact global maximizer (or to a consistent estimator) for non-convex cyclic ANMs with MNAR parameters

Circularity Check

0 steps flagged

No significant circularity; consistency theorem stated separately from algorithmic procedure

full rationale

The provided abstract and text present a consistency guarantee explicitly conditioned on exact maximization of the observed-data score in the large-sample limit. This is a standard asymptotic statement and does not reduce by construction to the EM-style alternating imputation/optimization steps actually implemented. No equations, self-citations, or fitted parameters are shown to be renamed as predictions or to define the target graph by tautology. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger populated at the level of stated modeling assumptions.

axioms (1)
  • domain assumption Data generated by additive noise model
    Framework integrates additive noise model with EM as stated in abstract.

pith-pipeline@v0.9.0 · 5679 in / 1055 out tokens · 20129 ms · 2026-05-23T19:12:56.656677+00:00 · methodology

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