📊 statistics

Bayesian weighted discrete-time dynamic models for association football prediction

This paper introduces a Bayesian weighted discrete-time dynamic modeling framework that utilizes adaptive shrinkage via spike and slab hyperpriors to flexibly track time-varying team abilities, demonstrating superior predictive performance over existing methods across major European football leagues and implemented in the open-source R package `footBayes`.

Original authors: Roberto Macrì-Demartino, Leonardo Egidi, Nicola Torelli

Published 2026-07-29

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Roberto Macrì-Demartino, Leonardo Egidi, Nicola Torelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future, but instead of looking at crystal balls, you are looking at a scoreboard. This is the world of sports analytics, a field where mathematicians and statisticians act like detectives, trying to figure out why a team wins or loses. For a long time, these detectives used a simple rule: they assumed a team's skill level was like a statue—stuck in one place, never changing. If a team was good at attacking last year, the models assumed they were equally good at attacking today, ignoring the fact that players get injured, coaches get fired, or star players get traded to other cities. But in the real world, teams are more like living, breathing organisms that grow, shrink, and change their minds every week. The big question for scientists is: How can we build a math model that is flexible enough to catch these sudden changes without getting confused?

This paper introduces a clever new way to predict football (soccer) matches using a method called "Bayesian weighted dynamic models." Think of it as a smart, adaptive memory system. In the past, models would either forget the past entirely or remember it too strictly, treating old data as if it were brand new. The authors, Roberto Macrì-Demartino, Leonardo Egidi, and Nicola Torelli, propose a system that uses "commensurate priors" with "spike-and-slab" logic. Imagine a librarian who is deciding how much to trust an old book about a team's skills. If the team is playing exactly as they did last month, the librarian puts a heavy "spike" on the old book, saying, "Trust this completely!" But if the team just got a new coach or lost their best striker, the librarian slides the book onto a "slab," saying, "This old info is probably wrong; let's look at what's happening right now." This allows the model to borrow strength from the past when it's safe to do so, but drop that past data instantly when things change.

The researchers tested this idea on six different types of mathematical models using data from the last five seasons of three major European leagues: the German Bundesliga, the English Premier League, and the Spanish La Liga. They split each season into two halves to catch mid-season changes, like winter transfer windows. When they compared their new "weighted" approach against older methods that assumed a constant rate of change, the results were clear. The new model consistently predicted match outcomes more accurately, especially in the final, high-pressure rounds of the season. For example, in the English Premier League, their best model (a Skellam model) achieved a "Brier score" of 0.545 for the final round, while in La Liga, their best model (a diagonal-inflated bivariate Poisson) hit a score of 0.462. Lower scores here mean better predictions.

The paper also looked at how the model tracked specific teams. It successfully spotted when a team like Bayern Munich had a slump or when Manchester United's attack got weaker over several years, adjusting its estimates much faster than the older models could. The authors found that by giving separate "weights" to a team's attacking and defensive skills, the model could handle situations where a team's offense might change quickly while their defense stays steady, or vice versa. They even made this tool available to the public as a free software package called footBayes. While the paper suggests this approach is a significant improvement, the authors are careful to note that it is a simulation-based finding and that future work could include even more data, like player injuries or market values, to make the predictions even sharper. Ultimately, this research suggests that treating a football team's ability as a flexible, changing story—rather than a static fact—leads to much better guesses about who will win the next game.

Technical Summary: Bayesian Weighted Discrete-Time Dynamic Models for Association Football Prediction

Problem Statement

The prediction of association football match outcomes has seen a proliferation of statistical models, yet a critical limitation persists in most existing approaches: the assumption that team-specific offensive and defensive abilities remain static over time. While some methods employ simple time-decay weighting or continuous-time state-space models, they often rely on a single, constant evolution precision shared across all teams and both ability types (attack and defense). This constraint fails to account for the heterogeneous nature of team performance evolution. Specifically, it cannot adequately handle periods of rapid change (e.g., transfer windows, managerial changes) where historical data should be downweighted, nor does it distinguish between the potentially different rates of change for offensive versus defensive capabilities. Furthermore, standard discrete-time dynamic models often assume a common evolution precision for all teams, ignoring that some teams may exhibit stable performance while others fluctuate significantly.

Methodology

The authors propose a Bayesian weighted discrete-time dynamic framework that integrates commensurate priors (Hobbs et al., 2011, 2012) into six standard goal-based models. The core innovation lies in the treatment of team abilities as time-varying parameters with adaptive, period-specific shrinkage.

1. Goal-Based Models

The study evaluates the proposed dynamic framework across six statistical distributions for goal counts:

Double Poisson (DP) and Bivariate Poisson (BP): To model goal counts and their potential dependence.
Negative Binomial (NB) and Skellam (SM): To address overdispersion and model goal differences directly.
Diagonally Inflated Bivariate Poisson (DIBP) and Zero-Inflated Skellam (ZISM): To correct for the underestimation of draw probabilities common in standard Poisson models.

2. Weighted Dynamic Priors

Unlike previous approaches (e.g., Owen, 2011) that use a random walk with a common precision $\sigma$ , the proposed model assigns separate, time-varying precision parameters ( $\phi_{att, \tau}$ and $\phi_{def, \tau}$ ) for each team's attack and defense abilities in each period $\tau$ .

Prior Structure: The ability in period $\tau$ is modeled as a normal distribution centered on the ability in period $\tau-1$ , with variance controlled by the precision $\phi_{k, \tau}$ .
Spike-and-Slab Hyperpriors: To determine the degree of information borrowing from the past, the precision parameters $\phi_{k, \tau}$ $ϕ_{k, τ}$ are assigned spike-and-slab hyperpriors.
- The spike component (high precision) encourages strong borrowing when current data aligns with historical trends, effectively shrinking the current estimate toward the past.
- The slab component (low precision) allows for high variability and minimal borrowing when current data conflicts with historical performance (e.g., due to transfers or coaching changes).
Adaptive Mechanism: This structure allows the model to automatically decide how much weight to give to past performance. If a team's performance is stable, the model borrows heavily; if a team undergoes a structural shift, the model downweights the past, allowing the current data to dominate.

3. Implementation

The models were implemented using the Stan probabilistic programming language via the R package footBayes. The study utilized data from the last five seasons (2020/2021–2024/2025) of the German Bundesliga, English Premier League (EPL), and Spanish La Liga. Each season was divided into two discrete periods (first and second halves) to capture midseason structural events.

Key Results

The performance of the proposed weighted dynamic approach was compared against two benchmarks: the discrete-time approach of Owen (2011) (common precision) and Egidi et al. (2018) (ability-specific but constant precision). Evaluation was conducted using Brier scores (lower is better) and Average of Correct Probabilities (ACP) (higher is better) across three scenarios: the entire second half of the season, the last three rounds, and the final round.

Predictive Accuracy: The weighted dynamic approach consistently yielded the lowest Brier scores and highest ACP values across all three leagues and all six goal-based models.
- In the Bundesliga, the weighted dynamic Bivariate Poisson model achieved a Brier score of 0.678 (vs. 0.683 for Egidi et al. and 0.687 for Owen) in the last three rounds.
- In the EPL, the weighted dynamic Diagonally Inflated Bivariate Poisson model achieved a Brier score of 0.602 and an ACP of 0.421, outperforming all other methods.
- In La Liga, the weighted dynamic approach achieved the best results with a Brier score of 0.499 and an ACP of 0.454.
Model Behavior: The analysis of team ability trajectories (e.g., Bayern Munich, Manchester United, Girona FC) demonstrated that the weighted dynamic model more accurately captured volatility and structural shifts.
- For stable teams, the model maintained accurate trend tracking.
- For teams with inconsistent performance (e.g., Girona's drop from 3rd to 16th, or Manchester United's decline), the model successfully identified rapid changes in attack and defense, whereas the benchmark models exhibited slower adaptation or failed to capture the crossover in ability strengths.
Computational Efficiency: Contrary to expectations that complex hyperpriors might increase computational cost, the weighted dynamic approach required less computation time to reach convergence compared to the Owen (2011) and Egidi et al. (2018) formulations.

Significance and Claims

The paper claims that the proposed methodology offers a more flexible and responsive framework for modeling team abilities in dynamic environments. By utilizing commensurate priors with spike-and-slab hyperparameters, the approach provides a formal mechanism for adaptive shrinkage, allowing the model to borrow historical information only to the extent justified by the data.

The authors assert that this leads to:

Superior Predictive Performance: Specifically in short-term forecasting scenarios (e.g., final rounds) where rapid adaptation to recent form changes is critical.
Realistic Trajectory Modeling: The ability to reflect midseason fluctuations and major transitions (transfers, coaching changes) more accurately than models assuming static or uniformly evolving parameters.
Generalizability: While applied to football, the framework is presented as a general solution for any domain involving sequential performance data with potential shifts in underlying ability (e.g., other sports).

The work is implemented in the open-source R package footBayes, facilitating reproducibility and further application in sports analytics.