# MissDAG: Causal Discovery in the Presence of Missing Data with Continuous Additive Noise Models

arXiv: https://arxiv.org/abs/2205.13869 · HTML: https://mlsanity.com/paper/2205.13869/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 88 references that are in the map.
- First posted on arXiv 27 May 2022.

## Builds on (most cited first, 28 of 28)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Missing Data Imputation using Optimal Transport](https://mlsanity.com/paper/2002.03860/) (2020)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [Graphical Models for Processing Missing Data](https://mlsanity.com/paper/1801.03583/) (2018)
- [A Tutorial on Learning With Bayesian Networks](https://mlsanity.com/paper/2002.00269/) (2020)
- [Differentiable Causal Discovery Under Unmeasured Confounding](https://mlsanity.com/paper/2010.06978/) (2020)
- [Ordering-Based Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/2105.06631/) (2021)
- [Learning linear structural equation models in polynomial time and sample complexity](https://mlsanity.com/paper/1707.04673/) (2017)
- [MIRACLE: Causally-Aware Imputation via Learning Missing Data Mechanisms](https://mlsanity.com/paper/2111.03187/) (2021)
- [Causal Discovery in the Presence of Missing Data](https://mlsanity.com/paper/1807.04010/) (2018)
- [Identifiable Generative Models for Missing Not at Random Data Imputation](https://mlsanity.com/paper/2110.14708/) (2021)
- [The Bayesian Structural EM Algorithm](https://mlsanity.com/paper/1301.7373/) (2013)
- [Full Law Identification In Graphical Models Of Missing Data: Completeness Results](https://mlsanity.com/paper/2004.04872/) (2020)
- [Identification In Missing Data Models Represented By Directed Acyclic Graphs](https://mlsanity.com/paper/1907.00241/) (2019)
- [Reliable Causal Discovery with Improved Exact Search and Weaker Assumptions](https://mlsanity.com/paper/2201.05666/) (2022)
- [MissDeepCausal: Causal Inference from Incomplete Data Using Deep Latent Variable Models](https://mlsanity.com/paper/2002.10837/) (2020)
- [Learning Bayesian Networks with Incomplete Data by Augmentation](https://mlsanity.com/paper/1608.07734/) (2016)

## Built on by (most cited first, 22 of 22)

- [Harnessing Out-Of-Distribution Examples via Augmenting Content and Style](https://mlsanity.com/paper/2207.03162/) (2022)
- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Gene Regulatory Network Inference in the Presence of Dropouts: a Causal View](https://mlsanity.com/paper/2403.15500/) (2024)
- [Optimal Transport for Structure Learning Under Missing Data](https://mlsanity.com/paper/2402.15255/) (2024)
- [Mitigating Label Noise on Graph via Topological Sample Selection](https://mlsanity.com/paper/2403.01942/) (2024)
- [COKE: Causal Discovery with Chronological Order and Expert Knowledge in High Proportion of Missing Manufacturing Data](https://mlsanity.com/paper/2407.12254/) (2024)
- [Identification of Causal Structure in the Presence of Missing Data with Additive Noise Model](https://mlsanity.com/paper/2312.12206/) (2023)
- [psiDAG: Projected Stochastic Approximation Iteration for DAG Structure Learning](https://mlsanity.com/paper/2410.23862/) (2024)
- [The Robustness of Differentiable Causal Discovery in Misspecified Scenarios](https://mlsanity.com/paper/2510.12503/) (2025)
- [On the Sparse DAG Structure Learning Based on Adaptive Lasso](https://mlsanity.com/paper/2209.02946/) (2022)
- [Deception by Omission: Using Adversarial Missingness to Poison Causal Structure Learning](https://mlsanity.com/paper/2305.20043/) (2023)
- [Revealing Multimodal Causality with Large Language Models](https://mlsanity.com/paper/2509.17784/) (2025)
- [Causal Discovery for Irregularly Time Series with Consistency Guarantees](https://mlsanity.com/paper/2507.03310/) (2025)
- [Learning Cyclic Causal Models from Incomplete Data](https://mlsanity.com/paper/2402.15625/) (2024)
- [Revisiting Differentiable Structure Learning: Inconsistency of ℓ1 Penalty and Beyond](https://mlsanity.com/paper/2410.18396/) (2024)
- [MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data](https://mlsanity.com/paper/2410.18918/) (2024)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [PAIR-CI: Calibrated Conditional Independence Testing for Causal Discovery with Incomplete Data](https://mlsanity.com/paper/2605.04838/) (2026)
- [Causal Modeling of Selection in Evolution](https://mlsanity.com/paper/2606.05689/) (2026)
- [ITSY: Causal Discovery From Irregular Time-Series Data](https://mlsanity.com/paper/2609.23516/) (2026)
- [Generalizing Identifiability Results in Causal Discovery from Complete Data to Missing Data Settings](https://mlsanity.com/paper/2610.19211/) (2026)

## Abstract (arXiv)

State-of-the-art causal discovery methods usually assume that the observational data is complete. However, the missing data problem is pervasive in many practical scenarios such as clinical trials, economics, and biology. One straightforward way to address the missing data problem is first to impute the data using off-the-shelf imputation methods and then apply existing causal discovery methods. However, such a two-step method may suffer from suboptimality, as the imputation algorithm may introduce bias for modeling the underlying data distribution. In this paper, we develop a general method, which we call MissDAG, to perform causal discovery from data with incomplete observations. Focusing mainly on the assumptions of ignorable missingness and the identifiable additive noise models (ANMs), MissDAG maximizes the expected likelihood of the visible part of observations under the expectation-maximization (EM) framework. In the E-step, in cases where computing the posterior distributions of parameters in closed-form is not feasible, Monte Carlo EM is leveraged to approximate the likelihood. In the M-step, MissDAG leverages the density transformation to model the noise distributions with simpler and specific formulations by virtue of the ANMs and uses a likelihood-based causal discovery algorithm with directed acyclic graph constraint. We demonstrate the flexibility of MissDAG for incorporating various causal discovery algorithms and its efficacy through extensive simulations and real data experiments.