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Learning concepts from natural high-dimensional data (e.g., images) holds potential

A.2 Proof of proposition 1 Let Pœ{B,D,E}, k be a valid kernel (assumptions of theorem 1) with K Inversion of conditional with Bayes rule gives: 'W œS As a complement, we now explicit the simple forms taken by the posterior limit graph in each case. A.3 Proof of theorem 2 We consider the following hierarchical model, for Nonetheless it can be simplified as we now show. We focus on finding the optimal eigenvectors first. Only the left term in (18) depends on R. The optimization problem for eigenvectors writes: min tr! Note that the identity permutation i.e. for i œ [n], (i) =i is optimal in this case as the ( We will choose this U in what follows as the sign of the axes do not influence the characterization of the final result in Z as a PCA embedding. Note that this solution is not unique if there are repeated eigenvalues.

Classical causal structure learning algorithms often assume no latent confounders. However, it is usually impossible to enumerate and measure all task-related variables in real-world scenarios. Neglecting latent confounders may lead to spurious correlations among observed variables.

Large width limits have been a recent focus of deep learning research: modulo computational practicalities, do wider networks outperform narrower ones? Answering this question has been challenging, as conventional networks gain representational power with width, potentially masking any negative effects.

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.