# Visually Communicating and Teaching Intuition for Influence Functions

arXiv: https://arxiv.org/abs/1810.03260 · HTML: https://mlsanity.com/paper/1810.03260/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Oct 2018.

## Built on by (most cited first, 19 of 19)

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- [Data-Driven Influence Functions for Optimization-Based Causal Inference](https://mlsanity.com/paper/2208.13701/) (2022)
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- [Weighted-average quantile regression](https://mlsanity.com/paper/2203.03032/) (2022)
- [A Double Machine Learning Approach for the Evaluation of COVID-19 Vaccine Effectiveness under the Test-Negative Design: Analysis of Qu'ebec Administrative Data](https://mlsanity.com/paper/2310.04578/) (2023)
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- [An Orthogonal Learner for Individualized Outcomes in Markov Decision Processes](https://mlsanity.com/paper/2509.26429/) (2025)
- [Weighted Spline-Expanded Networks with Distributional Balancing for Continuous Treatment Effects](https://mlsanity.com/paper/2609.33740/) (2026)

## Abstract (arXiv)

Estimators based on influence functions (IFs) have been shown to be effective in many settings, especially when combined with machine learning techniques. By focusing on estimating a specific target of interest (e.g., the average effect of a treatment), rather than on estimating the full underlying data generating distribution, IF-based estimators are often able to achieve asymptotically optimal mean-squared error. Still, many researchers find IF-based estimators to be opaque or overly technical, which makes their use less prevalent and their benefits less available. To help foster understanding and trust in IF-based estimators, we present tangible, visual illustrations of when and how IF-based estimators can outperform standard ``plug-in'' estimators. The figures we show are based on connections between IFs, gradients, linear approximations, and Newton-Raphson.