# Model-based Clustering with Missing Not At Random Data

arXiv: https://arxiv.org/abs/2112.10425 · HTML: https://mlsanity.com/paper/2112.10425/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 71 references that are in the map.
- First posted on arXiv 20 Dec 2021.

## Builds on (most cited first, 7 of 7)

- [On the consistency of supervised learning with missing values](https://mlsanity.com/paper/1902.06931/) (2019)
- [Imputation and low-rank estimation with Missing Not At Random data](https://mlsanity.com/paper/1812.11409/) (2018)
- [k-POD: A Method for k-Means Clustering of Missing Data](https://mlsanity.com/paper/1411.7013/) (2014)
- [Model Based Clustering for Mixed Data: clustMD](https://mlsanity.com/paper/1511.01720/) (2015)
- [Model-based clustering of Gaussian copulas for mixed data](https://mlsanity.com/paper/1405.1299/) (2014)
- [Handling missing data in model-based clustering](https://mlsanity.com/paper/2006.02954/) (2020)
- [Clustering Data with Nonignorable Missingness using Semi-Parametric Mixture Models](https://mlsanity.com/paper/2009.07662/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data](https://mlsanity.com/paper/2201.12020/) (2022)
- [Causal Clustering for Conditional Average Treatment Effects Estimation and Subgroup Discovery](https://mlsanity.com/paper/2509.05775/) (2025)

## Abstract (arXiv)

Model-based unsupervised learning, as any learning task, stalls as soon as missing data occurs. This is even more true when the missing data are informative, or said missing not at random (MNAR). In this paper, we propose model-based clustering algorithms designed to handle very general types of missing data, including MNAR data. To do so, we introduce a mixture model for different types of data (continuous, count, categorical and mixed) to jointly model the data distribution and the MNAR mechanism, remaining vigilant to the relative degrees of freedom of each. Several MNAR models are discussed, for which the cause of the missingness can depend on both the values of the missing variable themselves and on the class membership. However, we focus on a specific MNAR model, called MNARz, for which the missingness only depends on the class membership. We first underline its ease of estimation, by showing that the statistical inference can be carried out on the data matrix concatenated with the missing mask considering finally a standard MAR mechanism. Consequently, we propose to perform clustering using the Expectation Maximization algorithm, specially developed for this simplified reinterpretation. Finally, we assess the numerical performances of the proposed methods on synthetic data and on the real medical registry TraumaBase as well.