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Influence functions come from robust statistics and were adapted to deep learning by Koh and Liang in 2017. The core question is counterfactual: how would the model's loss on a test point change if a particular training example were removed or up-weighted? Rather than actually retraining (which is hopelessly expensive), influence functions approximate that change using calculus. They compute the gradient of the loss for the training point and the test point, then connect them through the inverse Hessian of the loss, which captures the curvature of the model's parameter space. Under Koh and Liang’s standard upweighting-on-test-loss convention, I_up,loss measures how infinitesimally increasing a training point’s weight changes the loss on a chosen test point. A positive value means upweighting raises test loss (a harmful influence); a negative value means it lowers test loss (a helpful influence). Removing a point reverses the sign of the first-order estimated change. Other influence scores may target predictions or use a different sign convention, so define the quantity before interpreting its sign. The result is a ranked list of the most responsible training examples.