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

Abstract

This study develops a hierarchical Bayesian framework that integrates expert domain knowledge to quantify player-specific effects in expected goals (xG) estimation, addressing a limitation of standard models that treat all players as identical finishers. Using 9,970 shots from StatsBomb's 2015-16 data and Football Manager 2017 ratings, we combine Bayesian logistic regression with informed priors to stabilise player-level estimates, especially for players with few shots. The hierarchical model reduces posterior uncertainty relative to weak priors and achieves strong external validity: hierarchical and baseline predictions correlate at R2 = 0.75, while an XGBoost benchmark validated against StatsBomb xG reaches R2 = 0.833. The model uncovers interpretable specialisation profiles, including one-on-one finishing (Aguero, Suarez, Belotti, Immobile, Martial), long-range shooting (Pogba), and first-touch execution (Insigne, Salah, Gameiro). It also identifies latent ability in underperforming players such as Immobile and Belotti. The framework supports counterfactual "what-if" analysis by reallocating shots between players under identical contexts. Case studies show that Sansone would generate +2.2 xG from Berardi's chances, driven largely by high-pressure situations, while Vardy-Giroud substitutions reveal strong asymmetry: replacing Vardy with Giroud results in a large decline (about -7 xG), whereas the reverse substitution has only a small effect (about -1 xG). This work provides an uncertainty-aware tool for player evaluation, recruitment, and tactical planning, and offers a general approach for domains where individual skill and contextual factors jointly shape performance.