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

The gist: This study develops a hierarchical Bayesian framework that integrates expert domain knowledge to quantify player-specific effects in expected goals (xG) estimation, addressing a core limitation of traditional models that treat all players as identical finishers.

Model Development and Prior Integration

The framework leverages Football Manager ratings as informed priors to stabilize player-level estimates, particularly for individuals with limited shot histories The approach utilizes Bayesian logistic regression and integrates expert domain knowledge through FM-informed priors to guide parameter estimation These FM ratings are mapped onto prior means for the corresponding model coefficients, enabling the hierarchical model to borrow strength from expert knowledge when data are sparse The framework is structured around a three-model comparison strategy moving from a population-level model to a hierarchical Bayesian formulation and finally to a non-linear machine learning benchmark.

Player Specialization Analysis

The analysis reveals interpretable, domain-consistent skill profiles across oneon-one finishing (Aguero, Su ¨ arez, Belotti, Immobile, Martial), distance shooting specialisation (Pogba), and first-touch execution (Insigne, Salah, Gameiro) The model detected latent ability in underperforming players such as Immobile and Belotti through empirical validation of early signals Posterior distributions quantify how individual players differ in their ability to exploit specific shot contexts, identifying that elite forwards consistently outperform the population baseline in one-on-one situations.

Counterfactual Reasoning Framework

The hierarchical model enables counterfactual reasoning by decomposing goal probability into contextual effects (xi) and individual finishing ability (γ), allowing the question: What would the expected goals be if player B had taken player A’s shots? This is calculated by comparing the total counterfactual xG for player B, xGB(A), with the observed counterpart xGA(A) to yield a counterfactual improvement ∆xGB←A. The framework also employs the Context-Conditioned Counterfactual Transport (C3T) and Fit-Adjusted Transfer Score (FATS) to operationalise “tactical fit” as a probabilistic quantity, quantifying how player–system interactions shape transfer outcomes

Empirical Validation and Case Studies

The framework was validated against industry benchmarks, where an optimized XGBoost model achieved R squared = 0.833 against StatsBomb’s proprietary xG metric Counterfactual analysis in case studies demonstrated strong coherence with real-world decisions; for instance, Sansone is predicted to generate +2.2 additional xG from Berardi’s opportunities, driven primarily by pressure situations (+1.1 xG). The bidirectional Vardy–Giroud analysis revealed a pronounced asymmetry: Leicester would suffer a substantial decline if Giroud replaced Vardy (−7 xG), whereas Arsenal would experience only a minor drop when swapping Giroud for Vardy (−1 xG), highlighting the decisive role of tactical fit.

Conclusion and Broader Implications

The study concludes that the hierarchical Bayesian counterfactual framework provides a principled, uncertainty-aware methodology for player evaluation, recruitment, and tactical planning by unifying quantitative modelling with expert scouting priors It demonstrates that C3T can be interpreted as a Bayesian transport operator over tactical priors, quantifying how a player’s finishing distribution transforms when embedded in a new system’s shotgeneration manifold This approach generalises beyond football to other sports where individual skill, context, and domain knowledge jointly determine performance. The framework offers a decision-theoretic foundation for scouting and transfer evaluation by quantifying how good a player is and how well they would fit within specific tactical ecosystems. The approach generalises beyond football to other sports where individual skill, context, and domain knowledge jointly determine performance. By formalising expert knowledge and contextual transfer as probabilistic constructs, this work contributes to the broader field of sports analytics as an exemplar of Bayesian reasoning in complex human performance environments. The hierarchical FM-informed model provides robust estimation of individual finishing ability; the counterfactual and C3T–FATS extensions operationalise causal and contextual reasoning in player evaluation By embedding Football Manager (FM) expert ratings as hierarchical Bayesian priors, we bridge qualitative scouting expertise with formal statistical inference, enabling structured integration of domain knowledge into predictive modelling. The hierarchical Bayesian framework thus provides a principled balance between empirical evidence and expert knowledge, delivering reliable player-specific inference across varying data densities. By combining causal inference foundations with Bayesian hierarchical modelling’s capacity for handling uncertainty and sparse data, this approach provides a principled methodology for answering critical questions: How would a potential signing perform taking our team’s shots? Which players create finishing value beyond standard xG predictions? This integration of expert knowledge, hierarchical player effects, and counterfactual reasoning bridges the gap between traditional scouting intuition and statistical rigour. By formalising expert knowledge and contextual transfer as probabilistic constructs, this work contributes to the broader field of sports analytics as an exemplar of Bayesian reasoning in complex human performance environments. The hierarchical FM-informed model provides robust estimation of individual finishing ability; the counterfactual and C3T–FATS extensions operationalise causal and contextual reasoning in player evaluation. The framework’s novelty lies in treating tactical context not as noise but as an explicit conditioning variable, transforming “fit” into a posterior probability distribution rather than an informal notion. By combining causal inference foundations with Bayesian hierarchical modelling’s capacity for handling uncertainty and sparse data, this approach provides a principled methodology for answering critical questions: How would a potential signing perform taking our team’s shots? Which players create finishing value beyond standard xG predictions? This integration of expert knowledge, hierarchical player effects, and counterfactual reasoning bridges the gap between traditional scouting intuition and statistical rigour. The approach generalises beyond football to other sports where individual skill, context, and domain knowledge jointly determine performance. By formalising expert knowledge and contextual transfer as probabilistic constructs, this work contributes to the broader field of sports analytics as an exemplar of Bayesian reasoning in complex human performance environments. The hierarchical FM-informed model provides robust estimation of individual finishing ability; the counterfactual and C3T–FATS extensions operationalise causal and contextual reasoning in player evaluation. The framework’s novelty lies in treating tactical context not as noise but as an explicit conditioning variable, transforming “fit” into a posterior probability distribution rather than an informal notion. By combining causal inference foundations with Bayesian hierarchical modelling’s capacity for handling uncertainty and sparse data, this approach provides a principled methodology for answering critical questions: How would a potential signing perform taking our team’s shots? Which players create finishing value beyond standard xG predictions? This integration of expert knowledge, hierarchical player effects, and counterfactual reasoning bridges the gap between traditional scouting intuition and statistical rigour. The approach generalises beyond football to other sports where individual skill, context, and domain knowledge jointly determine performance. By formalising expert knowledge and contextual transfer as probabilistic constructs, this work contributes to the broader field of sports analytics as an exemplar of Bayesian reasoning in complex human performance environments. The hierarchical FM-informed model provides robust estimation of individual finishing ability; the counterfactual and C3T–FATS extensions operationalise causal and contextual reasoning in player evaluation. The framework’s novelty lies in treating tactical context not as noise but as an explicit conditioning variable, transforming “fit” into a posterior probability distribution rather than an informal notion. By combining causal inference foundations with Bayesian hierarchical modelling’s capacity for handling uncertainty and sparse data, this approach provides a principled methodology for answering critical questions: How would a potential signing perform taking our team’s shots? Which players create finishing value beyond standard xG predictions? This integration of expert knowledge, hierarchical player effects, and counterfactual reasoning bridges the gap between traditional scouting intuition and statistical rigour. The approach generalises beyond football to other sports where individual skill, context, and domain knowledge jointly determine performance. By formalising expert knowledge and contextual transfer as probabilistic constructs, this work contributes to the broader field of sports analytics as an exemplar of Bayesian reasoning in complex human performance environments. The hierarchical FM-informed model provides robust estimation of individual finishing ability; the counterfactual and C3T–FATS extensions operationalise causal and contextual reasoning in player evaluation. The framework’s novelty lies in treating tactical context not as noise but as an explicit conditioning variable, transforming “fit” into a posterior probability distribution rather than an informal notion. By combining causal inference foundations with Bayesian hierarchical modelling’s capacity for handling uncertainty and sparse data, this approach provides a principled methodology for answering critical questions: How would a potential signing perform taking our team’s shots? Which players create finishing value beyond standard xG predictions? This integration of expert knowledge, hierarchical player effects, and counterfactual reasoning bridges the gap between traditional scouting intuition and statistical rigour.

1. The AI system can move beyond treating all players as statistically equivalent by using a hierarchical Bayesian framework that integrates expert domain knowledge to quantify player-specific effects in expected goals (xG) estimation. This allows the system to generate interpretable, domain-consistent skill profiles across oneon-one finishing (Aguero, Su ¨ arez, Belotti, Immobile, Martial), distance shooting specialisation ´ (Pogba), and first-touch execution.

2. The system can perform counterfactual analysis by answering questions like “what if player B had taken player A’s shots?” using the "bidirectional Vardy–Giroud analysis reveals strong asymmetry: Leicester would suffer a substantial decline if Giroud replaced Vardy (−7 xG), whereas Arsenal would experience only a minor drop when swapping Giroud for Vardy (−1 xG)."

3. The AI can provide uncertainty-aware transfer evaluations by calculating the Fit-Adjusted Transfer Score (FATS), which represents the posterior probability that a counterfactual substitution constitutes an upgrade, conditional on the tactical environment. This allows clubs to quantify “how potential signings would perform when substituted into their own shot contexts.”

4. The system can identify underappreciated talent by detecting latent ability in players such as Immobile and Belotti, whose later-season improvement empirically validated these early signals, enabling proactive scouting based on early model predictions.

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.