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Doubly Robust Machine Learning for Population Size Estimation with Missing Covariates: Application to Gaza Conflict Mortality

This paper develops a novel doubly robust machine learning framework to improve population size estimation when covariates are missing, demonstrating its effectiveness by re-estimating Gaza Strip mortality rates and finding that the death toll is approximately 26% higher than previously reported.

Original authors: Mateo Dulce Rubio, Edward H. Kennedy, Nicholas P. Jewell

Published 2026-02-11

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CC BY 4.0

Original authors: Mateo Dulce Rubio, Edward H. Kennedy, Nicholas P. Jewell

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 trying to count how many people are attending a massive, crowded music festival, but there’s a catch: you don't have a master list of every attendee. Instead, you only have three different "snapshots" taken at different times: one from the ticket scanners at the gates, one from the food court receipts, and one from the security camera footage.

This is exactly what scientists call "Capture-Recapture." By looking at how many people show up in two or three different lists, you can use math to guess how many people are actually there in total.

The Problem: The "Blurry" Data

The researchers in this paper are dealing with a much darker version of this problem. They are trying to estimate the number of deaths in the Gaza Strip during a conflict. In such chaotic settings, data is messy.

Imagine you have those three lists (Hospital records, Surveys, and Social Media posts), but there’s a major problem: The lists are incomplete.

For example, a hospital record might tell you a person was a male, but it might be missing their age. A social media post might mention a name but not the location. In statistics, these are "missing covariates."

Usually, scientists try to fix this by "filling in the blanks" (imputation)—basically, guessing the age based on other clues. But if your guesses are even slightly wrong, your final count of the total population could be wildly off. It’s like trying to solve a jigsaw puzzle where you’ve glued some of the pieces together incorrectly; the whole picture ends up looking wrong.

The Solution: The "Double-Check" Safety Net

The authors developed a new mathematical tool called "Doubly Robust Machine Learning."

Think of it like a high-tech GPS system for a driver in a fog.

The first layer (The Map): The system uses a "map" to guess where the roads are (this is the model of how people are captured in the lists).
The second layer (The Sensor): The system uses "sensors" to detect the actual road surface (this is the model of why information is missing).

The "Double Robustness" part is the magic: As long as at least one of these two layers is working correctly, the GPS will still get you to your destination accurately. Even if the "map" is a bit blurry or the "sensors" are a bit glitchy, the system corrects itself so the final answer isn't biased.

They also used Machine Learning, which acts like a super-fast assistant that can spot incredibly complex patterns in the data that a human or a simple calculator would miss.

The Result: A More Accurate (and Sobering) Truth

When the researchers applied this "Double-Check" method to the Gaza mortality data, they found something important.

Previous methods (which relied on "filling in the blanks") had estimated a certain number of deaths. However, this new, more robust method suggested the number was actually about 26% higher than what official statistics had shown.

While their estimate was slightly more "conservative" (meaning it was more precise and had a tighter margin of error), it revealed that the true scale of the tragedy was likely larger than previously thought because the old methods were being tripped up by the "missing pieces" of the data.

Why This Matters

This isn't just about math; it's about truth in chaos.

When we are trying to understand humanitarian crises, wars, or outbreaks of disease, the data is almost always broken. People are hard to reach, records are destroyed, and information is sensitive. This paper provides a "mathematical shield" that allows researchers to look through the fog of missing information and get as close to the real truth as possible.

Technical Summary: Doubly Robust Machine Learning for Population Size Estimation with Missing Covariates

1. Problem Statement

Population size estimation via capture-recapture (CR) methods is essential for studying hard-to-reach or vulnerable populations (e.g., victims of conflict, people using drugs). While classical models (like Lincoln-Petersen) assume homogeneous capture probabilities, modern approaches incorporate auxiliary covariates to account for individual heterogeneity.

However, a critical methodological gap exists: missing covariate information. In real-world settings, sensitive data is often withheld, records are incomplete, or probabilistic record linkage creates conflicting attributes. Existing solutions—such as complete-case analysis or a priori imputation—often introduce significant bias, especially when the missingness mechanism is related to both the covariates and the capture process.

2. Methodology

The authors propose a novel nonparametric estimation framework that integrates semiparametric efficiency theory with machine learning (ML) to handle missing covariates.

A. Identification Assumptions:
To identify the total population size ( $n$ ) from observed data, the paper relies on two key assumptions:

No Highest-Order Interaction (NHOI): Within a conditional log-linear model, the $K$ -way interaction term is assumed to be zero. This allows the model to extrapolate from observed capture patterns to the unobserved population without requiring full independence between all lists.
Missing at Random (MAR): The missingness indicator ( $R$ ) is conditionally independent of the missing covariates ( $X$ ), given the always-observed covariates ( $V$ ) and the capture profile ( $Y$ ). This allows the model to use observed data to "weight" the missing information.

B. Estimation Framework:
The authors derive the Efficient Influence Function (EIF) for the inverse capture probability ( $\psi^{-1}$ ). Using this, they develop two estimators:

Plug-in Estimator: A baseline approach that substitutes estimated nuisance functions into the identification formula. It is prone to first-order bias when using flexible ML models.
One-step ML Estimator (Doubly Robust): The primary contribution. It augments the plug-in estimator with a bias-correction term derived from the EIF. This estimator is doubly robust, meaning it remains consistent if either the missingness propensity model or the capture probability model is correctly specified.

C. Statistical Properties:
The framework provides finite-sample guarantees. Unlike many asymptotic methods, the authors prove that the one-step estimator is approximately normally distributed and achieves near-optimal efficiency for any sample size $N$ , provided the ML nuisance functions converge at a sufficient rate (e.g., $o_P(N^{-1/4})$ ).

3. Key Contributions

Theoretical Innovation: Established a new identification expression for capture probabilities under the combined NHOI and MAR assumptions.
Algorithmic Robustness: Developed a doubly robust, one-step ML estimator that mitigates the bias typically introduced by flexible machine learning models in capture-recapture settings.
Inference Validity: Provided a method for constructing approximately valid confidence intervals for both the capture probability and the total population size, even in moderate sample sizes.
Practical Tooling: Created a framework that accommodates high-dimensional or continuous covariates through data-adaptive ML methods (e.g., Random Forests).

4. Results

A. Simulation Studies:
The authors compared their method against naive imputation (using Random Forest to fill in missing values).

Bias/RMSE: As the missingness rate increased (up to 70%), imputation-based methods showed rapidly increasing bias and error. The MAR-based doubly robust estimators maintained consistently low bias and RMSE.
Coverage: Imputation-based confidence intervals failed significantly (dropping below 75% coverage), whereas the proposed method maintained near-nominal 95% coverage.

B. Application to Gaza Conflict Mortality:
The methodology was applied to re-estimate deaths in the Gaza Strip (Oct 7, 2023 – June 30, 2024) using three-list data (Hospital, Survey, and Social Media).

Findings: The doubly robust estimator yielded an estimate of 59,441 deaths (95% CI: 50,708–68,173).
Comparison: This estimate was approximately 10% lower than previous estimates (which used imputation) but still indicated that the true death toll exceeds official statistics by approximately 26%.

5. Significance

This paper provides a principled statistical bridge between high-dimensional machine learning and classical capture-recapture theory. By solving the problem of missing covariates, it offers a more reliable tool for humanitarian organizations, epidemiologists, and ecologists working with "noisy" or incomplete datasets. The ability to provide valid inference in conflict settings—where data is often fragmented and sensitive—is of profound practical and ethical importance.