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Graphical Models for Processing Missing Data
Karthika Mohan, Judea Pearl
TL;DR
Missing-data methods leave transparency, estimability, and testability insufficiently addressed, especially for MNAR settings. This paper reviews graphical-model approaches that encode missingness assumptions, determine recoverability, and derive estimators and tests. Its central result is query dependence: some properties are unrecoverable while others can be recovered, although particular MNAR joint distributions remain nonrecoverable.
Problem
Traditional missing-data frameworks leave key issues of transparency, estimability, and testability relatively unexplored, while common software largely assumes MCAR or MAR.
Method
The paper represents missingness with graphical models, reads conditional independencies from m-graphs, and develops graphical procedures for recoverability and model testing.
Results
Recoverability is query dependent: some properties are unrecoverable while others can be proven recoverable using estimation algorithms, including in broad MNAR classes.
Takeaways & Limitations
Graphical tools support transparent modeling of missingness and extend consistent-estimation analysis beyond MCAR and MAR to many MNAR problems.
Takeaways & Limitations
A causal query must be identifiable in the substantive model before it can be recoverable under missingness.
Abstract
from arXiv · showhide
This paper reviews recent advances in missing data research using graphical models to represent multivariate dependencies. We first examine the limitations of traditional frameworks from three different perspectives: \textit{transparency, estimability and testability}. We then show how procedures based on graphical models can overcome these limitations and provide meaningful performance guarantees even when data are Missing Not At Random (MNAR). In particular, we identify conditions that guarantee consistent estimation in broad categories of missing data problems, and derive procedures for implementing this estimation. Finally we derive testable implications for missing data models in both MAR (Missing At Random) and MNAR categories.
1 Introduction
The paper reviews missing-data research through graphical models, focusing on transparency, recoverability, and testability. It shows how graph structure can classify missingness, determine estimability, derive consistent estimators, and expose assumptions to empirical scrutiny.
- Graphical models organize missing-data analysis around transparency, recoverability, and testability.The paper explicitly identifies these three aspects as its main focus.
- Testability: The paper develops tests for conditional-independence claims and MAR models while identifying dependence claims that cannot be tested.These results address the limited availability of compatibility tests under missingness.
- Transparency: Most missing-data software assumes MCAR or MAR, so users must determine whether those mechanisms fit their data.When this assumption fails, existing software has no guarantee of lower bias than complete-case analysis.
- Transparency: Graphical models make missingness classification more transparent by encoding assumptions causally and allowing conditional independencies to be read from graph structure.The paper presents graphs as a way to model the reasons for missingness rather than rely on intuition or blind MAR assumptions.
- Recoverability: Recoverability asks whether a desired parameter has any consistent estimator under an assumed model and how to construct one when it does.The paper presents methods for deriving consistent estimators for statistical and causal parameters.
- Recoverability: Many MNAR problems permit consistent estimation of selected parameters even when imputing all missing values is impossible.The paper states that estimable parameters can often be identified directly from graph structure.
2 Graphical Models for Missing Data: Missingness Graphs (m-graphs)
Missingness graphs (m-graphs) represent causal and missingness processes with observed proxies and missingness mechanisms, making MCAR, MAR, and MNAR classifications mechanically inspectable. They provide a graphical language for expressing assumptions, conditional independencies, and the distinction between observed-data and underlying distributions.
- 2.1 Missingness Graphs: Notations and Terminology: A missingness mechanism masks a variable when its status equals 1 and reveals the underlying value when its status equals 0.The proxy therefore contains either the missingness symbol m or the variable’s true value.
- 2.1 Missingness Graphs: Notations and Terminology: M-graphs extend causal graphs by representing partially observed variables with observed proxies and missingness mechanisms.For each partially observed variable, the proxy is observed while the mechanism records whether the underlying value is concealed.
- 2.1 Missingness Graphs: Notations and Terminology: M-graphs distinguish the observed-data distribution from the underlying distribution and encode causal, latent-confounding, and conditional-independence relationships.Directed edges represent causal relationships, bidirected edges represent a shared unobserved parent, and independencies are read using d-separation.
- 2.2 Missingness Categories: MCAR, MAR, and MNAR correspond to distinct m-graph structures, so classification becomes mechanical once the variable relationships are specified.The paper identifies the three categories in Figure 1(b), (c), and (d), respectively.
- 2.2.1 Missing At Random: A Brief Discussion: Graphical models make missingness assumptions more transparent than Rubin’s framework, whose independence assumptions are difficult to assess and whose Rubin-MAR condition is untestable.The paper presents variable-based graphical criteria as a response to this opacity and lack of testability.
- 2.2.1 Missing At Random: A Brief Discussion: Graphical tools support explicit modeling of missingness and extend estimation analysis beyond MAR to MNAR problems.This motivates analyzing the details of the missingness process rather than blindly assuming MAR.
3 Recoverability
The paper defines recoverability as consistent estimation from incomplete data and develops graphical criteria and procedures for determining when target distributions remain estimable, including under MNAR.
- Efficiency: Listwise deletion uses only complete cases and can waste data and lower estimate quality, so efficiency should guide the choice among graph-licensed decompositions.Other procedures can use incomplete samples to estimate portions of the target distribution.
- Nonrecoverability: When an MNAR arrow directly links a variable to its missingness indicator, the corresponding joint distribution may be nonrecoverable regardless of sample size or imputation method.The income example illustrates this impossibility for P(I) when I causes its own missingness.
- MNAR Recovery: MNAR data can still permit consistent recovery of selected parameters, even when missingness depends strongly on partially observed variables.For Figure 3, the target P(Y | X, Z1, Z2) is recoverable from samples where all variables are observed.
- Sequential Factorization: Ordered factorizations recover target distributions when each factor satisfies the required conditional-independence criterion.Theorem 1 permits sequential recovery of factors that are conditionally independent of the relevant missingness indicators.
- Nonrecoverability: Joint-distribution recovery is forbidden when a partially observed variable is adjacent to, or collider-connected through observed variables with, its missingness indicator.Theorem 2 gives necessary and sufficient conditions for joint recovery when missingness indicators have no edges between them.
- Alternative Recovery: Some joint distributions remain recoverable through R-factorization and other graphical distributional constraints, even when ordinary conditional independencies are absent.The paper gives an observed-data expression for P(X, Y, Z) and notes recovery despite P(X, RY, RX) lacking independencies.
4 Testability Under Missingness
Graphical criteria distinguish missingness claims that can genuinely be tested from independencies imposed by the recovery procedure itself. The framework yields testable implications for MAR and MCAR, while some MNAR claims remain untestable and recoverability can differ between statistically indistinguishable models.
- Untestable claims: Some recovered-distribution independencies are imposed by construction and therefore cannot refute the missingness model.For X⊥⊥Rx, any partially observed dataset can be made compatible with the model.
- Untestable claims: Conditional independencies of the form X⊥⊥Rx|Z, W, Rz are untestable when {Z, X} are partially observed and W is fully observed.
- Testable claims: A d-separation is testable when missingness indicators for partially observed variables are in the separating set or can be added without spoiling separation.
- Testable claims: Testable claims can be rewritten using proxy variables and operationalized as equations involving only observed quantities.These equations can be tested from the observed-data distribution using finite-sample conditional-independence techniques.
- MAR and MCAR: MAR is testable if and only if at least two variables are partially observed, whereas MAR cannot be verified when its tests are satisfied.A rejected test can help locate a likely model error, such as a missing edge between A and RB.
- MAR and MCAR: MCAR is testable if and only if |Vm| + |Vo| ≥2, while two models can share untestable independencies yet differ in recoverability.In the cited comparison, P(XY) is recoverable in one model but not the other.
5 Conclusions
The paper uses graphical models to study transparency, recoverability, and testability in missing-data problems, including broad MNAR classes. Its analysis is query-dependent and nonparametric, and it supports model diagnosis and data-collection decisions.
- Graphical models expose conditional-independence implications while the paper charts estimability and testability under missingness, including broad MNAR problems.
- Recoverability depends on the target query: one property may be unrecoverable while another is recoverable through an appropriate estimation algorithm.
- The results assume nonparametric models and require no functional or distributional assumptions about the relationships involved.
- The paper questions the practical value of the MCAR, MAR, and MNAR taxonomy once a missingness model is explicitly represented graphically.
- Testability criteria can identify specific misspecifications, while recoverability analysis can guide auxiliary-variable measurement and flag variables whose imprecision may compromise recovery.
6 Appendix
The appendix illustrates how graphical-model factorizations recover joint distributions from incomplete data. It decomposes the target into factors, recovers each factor from observed proxies and missingness indicators, and uses do-calculus when ordinary factor recovery is insufficient.
- Recovery procedure: The target factorization includes complete-case terms and conditional missingness probabilities such as P(Rz = 0), P(Rw = 0|Z), and P(Rx = 0|Y, W).
- Recovery procedure: Joint-distribution recovery proceeds by factorizing the target and checking whether every resulting factor is recoverable.For the illustrated target, recoverability of all factors establishes recoverability of the joint distribution.
- Observed-data factors: Complete-case and missingness terms are rewritten using observed proxies and missingness indicators under the graph’s conditional independencies.
- Causal sub-queries: The factor P(Rx = 0|Y, W) requires converting a probabilistic sub-query into a causal query using do-calculus.The procedure then proves recoverability by showing that all factors in the resulting expression are recoverable.
- Causal sub-queries: The causal sub-query is expressed using observed-data distributions after applying do-calculus rules and the graph’s independence relations.