# Recursive Equations For Imputation Of Missing Not At Random Data With Sparse Pattern Support

arXiv: https://arxiv.org/abs/2507.16107 · HTML: https://mlsanity.com/paper/2507.16107/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 62 references that are in the map.
- First posted on arXiv 21 Jul 2025.

## Builds on (most cited first, 21 of 21)

- [Graphical Models for Processing Missing Data](https://mlsanity.com/paper/1801.03583/) (2018)
- [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)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [Causal Inference Under Interference And Network Uncertainty](https://mlsanity.com/paper/1907.00221/) (2019)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [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)
- [Auto-G-Computation of Causal Effects on a Network](https://mlsanity.com/paper/1709.01577/) (2017)
- [Greedy structure learning from data that contain systematic missing values](https://mlsanity.com/paper/2107.04184/) (2021)
- [Semiparametric Inference for Non-monotone Missing-Not-at-Random Data: the No Self-Censoring Model](https://mlsanity.com/paper/1909.01848/) (2019)
- [Causal inference, social networks, and chain graphs](https://mlsanity.com/paper/1812.04990/) (2018)
- [Itemwise conditionally independent nonresponse modeling for incomplete multivariate data](https://mlsanity.com/paper/1609.00656/) (2016)
- [Causal Inference With Outcome-Dependent Missingness And Self-Censoring](https://mlsanity.com/paper/2306.05511/) (2023)
- [Pattern graphs: a graphical approach to nonmonotone missing data](https://mlsanity.com/paper/2004.00744/) (2020)
- [On Testability and Goodness of Fit Tests in Missing Data Models](https://mlsanity.com/paper/2203.00132/) (2022)
- [Causal and Counterfactual Views of Missing Data Models](https://mlsanity.com/paper/2210.05558/) (2022)
- [Unifying Gaussian LWF and AMP Chain Graphs to Model Interference](https://mlsanity.com/paper/1811.04477/) (2018)
- [Adjustment Criteria for Recovering Causal Effects from Missing Data](https://mlsanity.com/paper/1907.01654/) (2019)
- [The Lauritzen-Chen Likelihood For Graphical Models](https://mlsanity.com/paper/2207.13794/) (2022)
- [Multiple Imputation Guided by Full Law and Target Law Identifiability under the Missing Not At Random Assumption](https://mlsanity.com/paper/2410.18688/) (2024)

## Built on by (most cited first, 1 of 1)

- [Markov Missing Graph: A Graphical Approach for Missing Data Imputation](https://mlsanity.com/paper/2509.03410/) (2025)

## Abstract (arXiv)

A common approach for handling missing values in data analysis pipelines is multiple imputation via software packages such as MICE (Van Buuren and Groothuis-Oudshoorn, 2011) and Amelia (Honaker et al., 2011). These packages typically assume the data are missing at random (MAR), and impose parametric or smoothing assumptions upon the imputing distributions in a way that allows imputation to proceed even if not all missingness patterns have support in the data. Such assumptions are unrealistic in practice, and induce model misspecification bias on any analysis performed after such imputation. In this paper, we provide a principled alternative. Specifically, we develop a new characterization for the full data law in graphical models of missing data. This characterization is constructive, is easily adapted for the calculation of imputation distributions for both MAR and MNAR (missing not at random) mechanisms, and is able to handle lack of support for certain patterns of missingness. We use this characterization to develop a new imputation algorithm -- Multivariate Imputation via Supported Pattern Recursion (MISPR) -- which uses Gibbs sampling, by analogy with the Multivariate Imputation with Chained Equations (MICE) algorithm, but which is consistent under both MAR and MNAR settings, and is able to handle missing data patterns with no support without imposing additional assumptions beyond those already imposed by the missing data model itself. In simulations, we show MISPR obtains comparable results to MICE when data are MAR, and superior, less biased results when data are MNAR. Our characterization and imputation algorithm based on it are a step towards making principled missing data methods more practical in applied settings, where the data are likely both MNAR and sufficiently high dimensional to yield missing data patterns with no support at available sample sizes.