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.
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