footBayes: Bayesian Football Prediction

• footBayes is a Bayesian approach that models football outcomes as posterior predictive distributions using sequential in-play event data.
• It employs ordered multinomial probit models with adaptive dynamic priors and spike-and-slab hyperpriors to capture evolving team strengths.
• The framework offers improved predictive accuracy and better calibration, outperforming traditional models with lower Brier scores.

footBayes is the name used in recent football analytics literature for Bayesian prediction systems that treat football outcomes as posterior predictive objects rather than as fixed point estimates. In one usage, footBayes denotes a Bayesian framework for minute-by-minute forecasting within a match, built around an ordered multinomial outcome and sequential event data such as goals, red cards, corners, crosses, fouls, and shots (Divekar et al., 2023). In a later usage, footBayes denotes a free and open source R package that implements Bayesian weighted discrete-time dynamic goal-based models in which teams’ attacking and defensive abilities evolve over time under adaptive commensurate priors with spike-and-slab hyperpriors (Macrì-Demartino et al., 7 Aug 2025). Across both usages, the defining feature is probabilistic forecasting with explicit uncertainty quantification.

1. Conceptual scope and nomenclature

The term footBayes is associated with two closely connected, but technically distinct, objects in the literature. The 2023 formulation introduces footBayes as a Bayesian methodology for real-time forecasting within soccer matches. Its central task is to forecast the final match outcome at each minute using all information observed up to that time, thereby addressing the gap between pre-match models and actual in-play match evolution (Divekar et al., 2023). The outcome is modeled as an ordered multinomial variable with three states: home win, draw, or home loss.

The 2025 formulation uses footBayes as the name of an R package for Bayesian dynamic modelling for football. Here the emphasis is not minute-by-minute event accumulation but goal-based prediction under time-varying team strengths. The package implements six standard goal-based models and a weighted discrete-time dynamic prior structure that allows attack and defense parameters to evolve adaptively across periods such as half-seasons (Macrì-Demartino et al., 7 Aug 2025).

A common misconception is to treat footBayes as a single immutable model. The published record instead shows that the name refers both to a within-match ordered multinomial probit framework and to a software environment for dynamic goal-based Bayesian models. This suggests that footBayes is best understood as a Bayesian football-prediction line of work rather than as one fixed likelihood specification.

2. Real-time within-match model

In the real-time formulation, the match outcome for the home team in game $i$ is encoded as $Y_i \in \{-1,0,1\}$, where $-1$ denotes home loss, $0$ draw, and $+1$ home win. The model introduces a latent variable $\Pi_i$ and two cutoffs $\delta_1,\delta_2$ such that

$$Y_i = \begin{cases} -1 & \text{if } \Pi_i < \delta_1 \ 0 & \text{if } \delta_1 \leq \Pi_i \leq \delta_2 \ +1 & \text{if } \Pi_i > \delta_2 . \end{cases}$$

This is an ordered multinomial probit construction, with the latent Gaussian layer providing a coherent route to posterior predictive probabilities for all three outcomes (Divekar et al., 2023).

The model distinguishes between time-invariant and time-varying covariates. Time-invariant covariates include starting team strengths, which remain fixed for the duration of a match. Time-varying covariates are sequential minute-by-minute event counts for both teams, including goals, cards, corners, crosses, fouls, and shots on or off goal. For time $t$,

$$\Pi_i^{(t)} = z_i^\top \boldsymbol{\gamma} + \sum_{k=1}^K x_{ik}^{(t)\top}\boldsymbol{\beta}_k^{(t)} + \epsilon_i^{(t)},$$

where $Y_i \in \{-1,0,1\}$0 is the vector of time-invariant covariates, $Y_i \in \{-1,0,1\}$1 is the vector of cumulative counts up to time $Y_i \in \{-1,0,1\}$2 for event type $Y_i \in \{-1,0,1\}$3, $Y_i \in \{-1,0,1\}$4 is the coefficient vector for time-invariant effects, $Y_i \in \{-1,0,1\}$5 is the time-varying coefficient vector for event $Y_i \in \{-1,0,1\}$6, and $Y_i \in \{-1,0,1\}$7. A separate model is constructed for each minute $Y_i \in \{-1,0,1\}$8 using all event history up to that minute, and the framework explicitly allows the impact of every event type to vary with time (Divekar et al., 2023).

The prior structure is conjugate. The coefficients $Y_i \in \{-1,0,1\}$9 have a multivariate normal prior, each $-1$0 has a multivariate normal prior with a structure that ensures temporal correlation in effects and independence between home and away events, the cutoffs $-1$1 have normal priors, and $-1$2 is fixed for identifiability. This conjugate construction is central to the model’s computational tractability in a real-time setting.

3. Posterior computation and predictive output

The real-time footBayes model is estimated by Gibbs sampling. For each time point $-1$3, posterior simulation alternates between four blocks: a latent-variable step in which $-1$4 is drawn from a truncated normal distribution consistent with the observed match outcome; a coefficient step in which the stacked parameter vector $-1$5 is sampled from a multivariate normal conditional posterior; a cutoff step using a Dirichlet-Beta-truncated mapping to sample $-1$6 while preserving their ordering; and repetition until convergence, assessed with the Geweke statistic, denoted GW (Divekar et al., 2023).

Posterior predictive forecasting is performed minute by minute. For a new match observed up to time $-1$7, the trained model yields a predictive distribution for the latent variable and thus predictive probabilities for win, draw, and loss: $-1$8 Because the procedure is Bayesian, the output is a full posterior predictive distribution rather than only a point classification. Credible intervals are therefore available for predicted probabilities at each match minute, which differentiates the method from deterministic competitors (Divekar et al., 2023).

This feature has practical significance beyond numerical forecasting accuracy. The model produces minute-by-minute win, draw, and loss probabilities with credible intervals and updates them as non-goal events accumulate, so forecast movements need not be driven solely by goals. In the reported case studies, this yields more nuanced temporal probability paths than models that react mainly to the scoreline.

4. Empirical evaluation, calibration, and robustness

The empirical evaluation of the real-time framework uses 3,040 English Premier League matches from 2008 to 2016, with minute-by-minute events for 8 types and team strengths derived from player ratings. Models are trained on 90% of matches and tested on the remaining 10%. Performance is assessed both in-sample and out-of-sample using F1-score and Brier score (Divekar et al., 2023).

The comparison set includes a Generalized Linear Model using standard probit regression, Support Vector Machines with linear and RBF kernels, and Random Forests. All competitor models are retrained for each time point using the same covariate information as footBayes. Under these conditions, footBayes outperforms all competitor models throughout the match, with especially visible gains for draws and losses and increasingly strong gains in later match stages as more event data accumulate. Its Brier scores are consistently the lowest, indicating better calibrated probabilistic predictions, and its F1-scores for wins and losses improve with time (Divekar et al., 2023).

Posterior analysis also yields interpretable time-varying covariate effects. Goals are the most influential covariate and their effect increases late in the game. Red cards have a strong negative effect, especially when they occur late. Crosses have a negative effect toward the endgame, confirming prior studies. Shots-on-goal and corners have limited net effect when goals are already included. The paper further reports that GLM, RF, and SVM tend to overfit to goal difference only and lack sensitivity to non-goal events and match flow (Divekar et al., 2023).

Robustness checks support the stability of these findings. Predictive accuracy is higher for Big 6 teams at home, but the method remains robust across team-quality groups. Leave-one-team-out tests show that excluding all matches for a team from the training data still allows the model to forecast that team’s matches well. Goal-margin analyses show the lowest Brier scores for matches with large, nonzero goal differences, while draws remain challenging; nevertheless, footBayes still leads the competing methods (Divekar et al., 2023).

5. Dynamic goal-based models and the footBayes R package

The later footBayes package generalizes the name toward a reusable software environment for Bayesian football prediction. It implements six standard goal-based models: double Poisson, bivariate Poisson, diagonal-inflated bivariate Poisson, negative binomial, Skellam, and zero-inflated Skellam. The underlying scoring rates for home and away teams are parameterized as

$-1$9

The package is described as free and open source, interoperable via R, with MCMC sampling performed via Stan (Macrì-Demartino et al., 7 Aug 2025).

Its principal methodological contribution is a weighted discrete-time dynamic prior for attacking and defensive abilities. Instead of using a single common evolution precision for all teams and periods, footBayes assigns a period-specific, ability-specific precision parameter through commensurate priors: $0$0 with $0$1. High $0$2 implies strong shrinkage toward the previous period, whereas low $0$3 allows rapid adaptation to new data. Each $0$4 is itself given a spike-and-slab hyperprior based on a mixture of half-normal distributions, enabling either strong or weak borrowing according to the data. Identifiability is enforced by zero-sum constraints within each period: $0$5 This construction is intended to accommodate both gradual and abrupt changes in team strength, including changes associated with transfer windows or coaching changes (Macrì-Demartino et al., 7 Aug 2025).

Implementation is through Stan-based MCMC in R, with a typical setup of 4 chains, 2000 iterations each, and 1000 burn-in. The package supports standard dynamic and static goal-based models and allows users to specify model type, data, prior settings, and period definitions. Reported diagnostics include $0$6 and high ESS, alongside faster convergence and reliable stability (Macrì-Demartino et al., 7 Aug 2025).

Empirical evaluation uses the last five seasons from the German Bundesliga, English Premier League, and Spanish La Liga. The weighted dynamic approach is compared with the Owen (2011) model, which uses a single evolution precision, and the Egidi et al. (2018) model, which uses attack- and defense-specific but constant precision. Prediction tasks include forecasting the entire second half of the last season, the last three rounds, and the final round. Metrics reported are Brier Score, Average Correct Probability, Ranked Probability Score, and Pseudo-$0$7. Across leagues and supported models, footBayes yields better predictive performance than the other discrete-time dynamic models. The paper also reports lower computational time, with reductions of 32%–55%, and gives an example in which the diagonal-inflated bivariate Poisson model in La Liga attains a Brier score of 0.499 for footBayes versus 0.518 for Egidi and 0.520 for Owen on the last three rounds (Macrì-Demartino et al., 7 Aug 2025).

6. Relation to broader Bayesian football modeling

footBayes belongs to a broader Bayesian tradition in football prediction, but its distinctive contribution is the combination of posterior predictive forecasting with either sequential in-play covariates or adaptive discrete-time evolution of team abilities. Earlier work on football prediction had already established several nearby Bayesian paradigms. A cumulative probit model for UEFA Champions League matches treated home win, draw, and home loss through a latent Gaussian variable and recursively updated priors across seasons using external ratings such as UEFACR or FCWR (Foulley, 2015). Bayesian multinomial-Dirichlet models showed that simple count-based categorical forecasting could be competitive with more complex alternatives while remaining well calibrated (Diniz et al., 2017). A hierarchical Bayesian Poisson model combined historical data and bookmakers’ odds by modeling scoring rates as convex combinations of history-based and odds-derived parameters (Egidi et al., 2018). A separate Bayesian framework inferred player abilities from event counts and used those inferred abilities to extend a hierarchical team scoring model (Whitaker et al., 2017).

This context helps clarify what footBayes is and is not. It is not the first Bayesian football model, nor is it confined to one likelihood family. Rather, the literature shows two main specializations under the same name. One specialization addresses within-match forecasting by combining ordered multinomial probit regression with minute-by-minute event histories and time-varying event effects (Divekar et al., 2023). The other specialization addresses dynamic goal-based prediction by packaging multiple score models inside a Stan-based framework with period-specific adaptive shrinkage (Macrì-Demartino et al., 7 Aug 2025). This suggests that the unifying idea behind footBayes is methodological: Bayesian updating, interpretable latent structure, and uncertainty-aware football prediction.

From that perspective, footBayes occupies a specific niche within sports analytics. It differs from static pre-match models by updating in response to in-play information or evolving team strengths; it differs from purely discriminative machine-learning baselines by producing full posterior predictive distributions; and it differs from simpler Bayesian football models by emphasizing either fine-grained sequential events or weighted discrete-time dynamics. Its significance in the literature lies less in a single canonical equation than in the consolidation of Bayesian forecasting workflows for football around posterior inference, calibration, and interpretable latent effects (Divekar et al., 2023, Macrì-Demartino et al., 7 Aug 2025).