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Weighting-Based Identification and Estimation in Graphical Models of Missing Data

This paper proposes a tree-based identification algorithm and a corresponding recursive inverse probability weighting procedure to address the propagation of selection bias in graphical models of missing data, providing a constructive way to estimate complete data distributions.

Original authors: Anna Guo, Razieh Nabi

Published 2026-02-12

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Anna Guo, Razieh Nabi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about a group of people, but there is a major problem: the witnesses are selectively disappearing.

Some people leave the room before they can answer certain questions. Some people only answer questions about their favorite hobbies but refuse to talk about their income. This isn't random; it’s a pattern. If you only listen to the people who stayed, you’ll get a skewed, biased version of the truth.

This paper, written by Anna Guo and Razieh Nabi, provides a high-tech "detective toolkit" to solve this exact problem.

The Problem: The "Selective Witness" Bias

In statistics, we call this MNAR (Missing Not At Random).

Think of a survey about health. People who are feeling very sick might be too tired to finish the survey. If you only analyze the people who completed it, your data will make it look like everyone is perfectly healthy. You’ve fallen into a "selection bias" trap.

Standard tools (like "imputation," which is basically guessing what the missing people would have said) often fail here because they assume the missingness is random. But in the real world, the reason people are missing is often tied to the very thing you are trying to measure.

The Solution: The "Intervention Tree"

The authors propose a new way to look at this using Graphical Models (think of these as "maps of influence"). They don't just look at the data; they look at the map of why people are leaving.

To solve the mystery, they use a brilliant concept called an Intervention Tree.

The Analogy: The Master Puppeteer
Imagine the missingness is controlled by a puppeteer. Every time a witness disappears, a string is pulled. The authors realize that if they can understand the "logic" of the puppeteer—which strings lead to which disappearances—they can mathematically "undo" the puppeteer's actions.

However, there’s a catch: The "Butterfly Effect" of Interventions.
If you try to "force" a witness to stay (an intervention), you might accidentally trigger a chain reaction. For example, if you force a witness to talk about their income, they might get annoyed and leave the room entirely! This is what the authors call "Selection Bias Propagation."

Their algorithm acts like a master strategist. It builds a "tree" of possible moves. It asks:

"If I intervene here, will it cause a chain reaction that ruins my data elsewhere?"
"Is there a specific sequence of 'moves' I can make to isolate the truth without triggering a massive exit of witnesses?"

If the algorithm finds a path through the tree that avoids these chain reactions, it tells you: "Yes, the truth is identifiable!" It then gives you a mathematical formula (a "weighting" method) to re-balance the data, effectively giving more "volume" to the voices of the people who would have been there if the puppeteer hadn't pulled the strings.

Why This Matters

The paper isn't just theoretical; they proved it works through simulations and real-world data (like a survey of Finnish university students).

In short:

Old way: "Some people didn't answer. Let's guess what they would have said based on the people who did." (Often wrong).
The New way: "Let's map out the logic of why people are leaving, find a way to mathematically 'neutralize' that logic, and reconstruct the truth without being fooled by the pattern of disappearance."

It’s the difference between looking at a half-empty puzzle and using a mathematical lens to see exactly what the missing pieces must have looked like.

Technical Summary: Weighting-Based Identification and Estimation in Graphical Models of Missing Data

Authors: Anna Guo and Razieh Nabi (Emory University)
Date: February 12, 2026

1. The Problem: The Challenge of MNAR Data

In statistical inference, missing data is often categorized as Missing Completely at Random (MCAR) or Missing at Random (MAR). However, in many real-world scenarios, data are Missing Not at Random (MNAR), meaning the probability of a value being missing depends on the unobserved value itself (e.g., self-censoring).

The fundamental difficulty with MNAR data is non-identifiability: the underlying complete data distribution cannot generally be recovered from the observed data distribution without additional structural assumptions. While existing methods rely on parametric assumptions or sensitivity analyses, they often lack a systematic way to determine exactly what can be identified within a specific graphical structure.

2. Methodology: An Interventionist Graphical Framework

The authors propose a framework that treats missingness indicators ( $R$ ) as variables subject to interventions. They model the missingness mechanism using a conditional Directed Acyclic Graph (mDAG), while leaving the target data distribution ( $X$ ) unrestricted.

Key Methodological Concepts:

Interventionist Perspective: Instead of viewing missingness as a passive process, the authors treat the act of "observing" a variable as an intervention ( $R_k = 1$ ). Identification is achieved by finding a sequence of interventions on missingness indicators that "breaks" the selection bias.
Selection Bias and Propagation: The paper identifies two types of selection:
1. Counterfactual-induced selection: Occurs when a parent of a missingness indicator is itself missing.
2. Indicator-induced selection: Occurs when intervening on one indicator to fix a bias induces new bias in another.
Causal Irrelevancy (Autonomy): The authors leverage the property that a propensity score $\pi_k$ is invariant to interventions on other missingness indicators. This allows them to "intervene" on descendants of $R_k$ to sever problematic dependencies.

3. Key Contributions

A. The Tree-Based Identification Algorithm
The central innovation is a constructive algorithm that produces a forest of identification trees.

Functionality: For each missingness indicator, the algorithm searches for an "admissible" sequence of interventions.
Pruning Mechanism: A unique "tree-pruning" step is introduced. If an intervention intended to fix one bias induces a "propagating" selection bias that obstructs the identification of a subsequent variable, the algorithm prunes that branch of the intervention strategy.
Diagnostic Output: Unlike previous methods that provide a binary "identifiable/not identifiable" answer, this algorithm provides a diagnostic tool. It tells the researcher exactly which propensity scores are identifiable and which specific subsets of data (evaluation sets) are required to recover them.

B. Recursive Inverse Probability Weighting (IPW) Estimators
Building on the identification trees, the authors develop a new class of estimators:

Recursive IPW: These estimators mirror the intervention logic. If the identification of a variable requires intervening on a set of indicators, the estimator uses the product of their propensity scores as weights.
Estimating Equations: They derive valid estimating equations for both the missingness mechanism and target functionals (like means, regression coefficients, or causal effects).
Closure Construction: They introduce a "closure" operation to determine the minimal set of missingness indicators required to estimate a specific parameter, ensuring the estimator is mathematically sound.

4. Results and Empirical Validation

A. Simulation Studies
The authors tested the method against standard approaches (Complete Case Analysis, Amelia/EM, and MICE) across three tasks:

Mean Estimation: In MNAR settings, the proposed method consistently recovered the true mean, while imputation-based methods (Amelia, MICE) exhibited significant, non-attenuating bias.
Parametric Regression: The proposed method maintained near-nominal Type I error and correct coverage, whereas competing methods suffered from inflated error rates due to spurious associations.
Causal Effect Estimation: The method successfully estimated average causal effects in complex MNAR mDAGs where other methods failed.

B. Real-Data Application
The method was applied to the Finnish Student Feedback Survey. By modeling the missingness of study financing variables via an mDAG, the authors were able to recover target distribution parameters that closely matched maximum likelihood results, demonstrating practical utility in survey statistics.

5. Significance and Impact

This paper shifts the paradigm of MNAR analysis from "imputing values" to "identifying structures."

Theoretical Significance: It provides a rigorous, constructive bridge between causal graphical models and missing data theory, specifically addressing the "selection bias propagation" problem that previous causal frameworks overlooked.
Practical Significance: The accompanying R package, flexMissing, allows practitioners to input a suspected missingness graph and receive a mathematically grounded estimation strategy. It provides a way to perform valid inference in MNAR settings without making restrictive parametric assumptions about the target data.