Which Influence Are We Estimating? The Role of Counterfactual Specifications in Data Attribution

Zhe Li, Wei Zhao, Peixin Zhang, Jun Sun

Estimating the influence of training examples on model behavior is essential for data debugging, valuation, and attribution. Existing influence estimators often produce incompatible rankings, which are commonly ascribed to approximation error. We argue that a more fundamental source of disagreement is specification mismatch: influence depends on the behavior being attributed, the intervention applied to each training example, and the counterfactual training process that maps the intervention to a model response. These choices are especially important when the target behavior requires a tractable surrogate, such as query loss, a logit, or a margin. We formalize influence as a counterfactual estimand, distinguish specification mismatch across estimands from approximation error in estimating a fixed estimand, and organize representative estimators by their implied specifications. We further derive a local decomposition that exposes how behavior signals, training signals, and counterfactual parameter responses interact. Controlled experiments show that exact estimands under different specifications can induce different rankings, whereas approximation error grows as perturbations move…