# MIRACLE: Causally-Aware Imputation via Learning Missing Data Mechanisms

arXiv: https://arxiv.org/abs/2111.03187 · HTML: https://mlsanity.com/paper/2111.03187/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 47 references that are in the map.
- First posted on arXiv 4 Nov 2021.

## Builds on (most cited first, 14 of 14)

- [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)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [Handling Missing Data with Graph Representation Learning](https://mlsanity.com/paper/2010.16418/) (2020)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Missing Data Imputation using Optimal Transport](https://mlsanity.com/paper/2002.03860/) (2020)
- [Causal Discovery in the Presence of Missing Data](https://mlsanity.com/paper/1807.04010/) (2018)
- [CASTLE: Regularization via Auxiliary Causal Graph Discovery](https://mlsanity.com/paper/2009.13180/) (2020)
- [Full Law Identification In Graphical Models Of Missing Data: Completeness Results](https://mlsanity.com/paper/2004.04872/) (2020)
- [MIDA: Multiple Imputation using Denoising Autoencoders](https://mlsanity.com/paper/1705.02737/) (2017)
- [Identification In Missing Data Models Represented By Directed Acyclic Graphs](https://mlsanity.com/paper/1907.00241/) (2019)
- [What Is Meant by "Missing at Random"?](https://mlsanity.com/paper/1306.2812/) (2013)
- [Learning Dynamics of Linear Denoising Autoencoders](https://mlsanity.com/paper/1806.05413/) (2018)
- [Provable Bounds for Learning Some Deep Representations](https://mlsanity.com/paper/1310.6343/) (2013)

## Built on by (most cited first, 30 of 51)

- [TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second](https://mlsanity.com/paper/2207.01848/) (2022)
- [HyperImpute: Generalized Iterative Imputation with Automatic Model Selection](https://mlsanity.com/paper/2206.07769/) (2022)
- [Deep Learning for Multivariate Time Series Imputation: A Survey](https://mlsanity.com/paper/2402.04059/) (2024)
- [MedFuse: Multi-modal fusion with clinical time-series data and chest X-ray images](https://mlsanity.com/paper/2207.07027/) (2022)
- [ReMasker: Imputing Tabular Data with Masked Autoencoding](https://mlsanity.com/paper/2309.13793/) (2023)
- [Transformed Distribution Matching for Missing Value Imputation](https://mlsanity.com/paper/2302.10363/) (2023)
- [LLM-Forest: Ensemble Learning of LLMs with Graph-Augmented Prompts for Data Imputation](https://mlsanity.com/paper/2410.21520/) (2024)
- [MissDAG: Causal Discovery in the Presence of Missing Data with Continuous Additive Noise Models](https://mlsanity.com/paper/2205.13869/) (2022)
- [Causal Deep Learning](https://mlsanity.com/paper/2303.02186/) (2023)
- [The Missing Indicator Method: From Low to High Dimensions](https://mlsanity.com/paper/2211.09259/) (2022)
- [DiffPuter: Empowering Diffusion Models for Missing Data Imputation](https://mlsanity.com/paper/2405.20690/) (2024)
- [Self-Supervision Improves Diffusion Models for Tabular Data Imputation](https://mlsanity.com/paper/2407.18013/) (2024)
- [Data Imputation with Iterative Graph Reconstruction](https://mlsanity.com/paper/2212.02810/) (2022)
- [Diffusion Models for Tabular Data Imputation and Synthetic Data Generation](https://mlsanity.com/paper/2407.02549/) (2024)
- [Handling Missing Data via Max-Entropy Regularized Graph Autoencoder](https://mlsanity.com/paper/2211.16771/) (2022)
- [To Impute or not to Impute? Missing Data in Treatment Effect Estimation](https://mlsanity.com/paper/2202.02096/) (2022)
- [MM-Align: Learning Optimal Transport-based Alignment Dynamics for Fast and Accurate Inference on Missing Modality Sequences](https://mlsanity.com/paper/2210.12798/) (2022)
- [Optimal Transport for Structure Learning Under Missing Data](https://mlsanity.com/paper/2402.15255/) (2024)
- [Knockout: A simple way to handle missing inputs](https://mlsanity.com/paper/2405.20448/) (2024)
- [In-Database Data Imputation](https://mlsanity.com/paper/2401.03359/) (2024)
- [Rethinking the Diffusion Models for Numerical Tabular Data Imputation from the Perspective of Wasserstein Gradient Flow](https://mlsanity.com/paper/2406.15762/) (2024)
- [Incomplete Graph Learning: A Comprehensive Survey](https://mlsanity.com/paper/2502.12412/) (2025)
- [Adversarial Learning for Feature Shift Detection and Correction](https://mlsanity.com/paper/2312.04546/) (2023)
- [Impute-MACFM: Imputation based on Mask-Aware Flow Matching](https://mlsanity.com/paper/2509.23126/) (2025)
- [Navigating causal deep learning](https://mlsanity.com/paper/2212.00911/) (2022)
- [Data Imputation from the Perspective of Graph Dirichlet Energy](https://mlsanity.com/paper/2304.04474/) (2023)
- [Learning Cyclic Causal Models from Incomplete Data](https://mlsanity.com/paper/2402.15625/) (2024)
- [New User Event Prediction Through the Lens of Causal Inference](https://mlsanity.com/paper/2407.05625/) (2024)
- [MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data](https://mlsanity.com/paper/2410.18918/) (2024)
- [Imputation-free Learning of Tabular Data with Missing Values using Incremental Feature Partitions in Transformer](https://mlsanity.com/paper/2504.14610/) (2025)

## Abstract (arXiv)

Missing data is an important problem in machine learning practice. Starting from the premise that imputation methods should preserve the causal structure of the data, we develop a regularization scheme that encourages any baseline imputation method to be causally consistent with the underlying data generating mechanism. Our proposal is a causally-aware imputation algorithm (MIRACLE). MIRACLE iteratively refines the imputation of a baseline by simultaneously modeling the missingness generating mechanism, encouraging imputation to be consistent with the causal structure of the data. We conduct extensive experiments on synthetic and a variety of publicly available datasets to show that MIRACLE is able to consistently improve imputation over a variety of benchmark methods across all three missingness scenarios: at random, completely at random, and not at random.