What If They Took the Shot? A Hierarchical Bayesian Framework for Counterfactual Expected Goals

Abstract

How would the probability of scoring change if a different player took the same shot? Although expected goals models describe the difficulty of scoring opportunities, comparing players on identical chances requires accounting for uncertainty in their estimated conversion probabilities. We address this question through a hierarchical Bayesian framework that evaluates alternative shooters under the same observed circumstances. Using 35,700 eligible shots from four leagues during the 2015/16 season, we compare a contextual benchmark with a primary player-intercept model and a provisional extension that allows player differences to vary with shooting conditions. Paired posterior draws preserve shared uncertainty when individual probability differences are aggregated across opportunities. Chronological validation demonstrates improved predictive performance relative to a constant-rate baseline, but introducing player effects provides negligible additional gains. Controlled single-seed experiments show that greater information per player improves estimation accuracy and narrows uncertainty intervals, although distinguishing small differences remains difficult and confident conclusions about near-null effects can become more frequent. Applications involving Giroud--Vardy and Berardi--Sansone similarly yield estimated differences that are small relative to their posterior uncertainty. These findings highlight an important distinction between predicting the outcome of a shot and establishing whether one player would be more likely to convert it. The proposed framework makes this distinction explicit by evaluating alternative shooters on fixed recorded opportunities while accounting for their contextual exposure. The resulting comparisons concern conditional conversion rather than the causal consequences of replacing a player in a match.