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A Bayesian bivariate conditional Poisson regression for goal dependence in the English Premier League

This paper introduces a Bayesian bivariate Conditional Poisson regression model to analyze English Premier League data, revealing a negative correlation between home and away goal counts and demonstrating how match attendance and fouls asymmetrically influence scoring dynamics.

Original authors: Marcus Nolan, Wagner Barreto-Souza, Luiza S. C. Piancastelli, Raanju R. Sundararajan

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Marcus Nolan, Wagner Barreto-Souza, Luiza S. C. Piancastelli, Raanju R. Sundararajan

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 watching a football match. The score isn't just a random number generator spitting out goals; it's a story of two teams dancing, pushing, and reacting to each other. If the home team scores, the away team might panic and attack wildly, or they might sit back and try to protect their lead. This means the number of goals one team scores is likely connected to how many the other team scores. For decades, statisticians have tried to write math equations to predict these scores. The old way of thinking was like rolling two separate dice: one for the home team, one for the away team, assuming what happens to one has absolutely nothing to do with the other. But anyone who has watched a game knows that's not true. The teams are in the same room, reacting to the same crowd, the same referee, and each other's moves.

This paper dives into that messy, exciting connection. The authors wanted to build a better "crystal ball" for football scores that doesn't just guess the numbers but understands the relationship between them. They used a fancy new type of math called a "Bayesian bivariate conditional Poisson regression." Don't let the big words scare you. Think of "Poisson" as a way to count things that happen randomly over time, like goals. "Bivariate" just means they are looking at two things at once (home goals and away goals). "Conditional" is the magic part: it means they are asking, "If the home team scores this many, what does that tell us about how many the away team will score?" Finally, "Bayesian" is a way of using math to update our guesses as we learn more, giving us a clear picture of how sure we can be about our predictions. Why does this matter? Because understanding how teams interact helps us see the hidden rules of the game, like how a roaring crowd might make the home team play better, or how a lead changes the whole game's strategy.

The New Game Plan: A Two-Way Street

The researchers took a fresh look at 1,140 matches from the English Premier League (EPL). They picked three specific seasons to get a perfect mix of situations: the 2018–19 season (normal crowds), the 2020–21 season (almost no crowds because of the pandemic), and the 2023–24 season (crowds back again). This gave them a natural experiment to see how the presence of fans changes the game. They also tracked how many fouls each team committed, treating fouls like a measure of how hard the teams were trying to stop each other.

Instead of the old "two separate dice" method, the authors built a new model where the goals are linked. They tested two directions for this link: Does the home team's score influence the away team's score (Home → Away), or does the away team's score influence the home team's score (Away → Home)?

The Big Discovery: The "Bus Parking" Effect

The most surprising finding is that the relationship between the two teams' scores is actually negative. In the world of football math, this is a big deal. Most old models assumed that if one team scored a lot, the other probably did too (positive correlation), or that they were totally unrelated. But this paper found that when the home team scores, the away team tends to score fewer goals.

The authors suggest this is like a tactical game of "parking the bus." When the home team gets a goal, the away team might get frustrated or scared, or the home team might decide to sit back and defend their lead so tightly that the away team can't get any shots off. The math showed a clear signal: for every extra goal the home team scores, the expected number of goals for the away team drops by about 10%. This negative link was strong enough that the authors are confident it's a real pattern, not just a fluke.

The Crowd's One-Sided Cheer

The study also looked at how the crowd affects the game. They found a very clear "home advantage" asymmetry. When the stadium was full (around 38,000 people), the home team scored more goals. The math showed that a 1% increase in attendance was linked to a tiny but real increase in home goals.

However, the crowd didn't seem to help the away team at all. Even when the stadium was packed, the number of goals the away team scored didn't go up. It was as if the crowd was a special boost only for the team playing on their own turf. When the fans were gone (during the 2020–21 season with only 462 people on average), the home team's scoring dropped, and the away team's scoring actually went up slightly, making the game more balanced. This confirms that the crowd is a secret weapon for the home team, but it doesn't give the visiting team a boost.

What About the Fouls?

The researchers also checked if the number of fouls committed by one team helped the other team score. The results here were a bit fuzzy. The data suggested that when the away team fouled a lot, the home team might score a few more goals, but the evidence wasn't strong enough to be 100% sure. It's possible that fouls are just a sign of a messy game, but the paper couldn't say for certain that fouling directly causes more goals for the opponent.

The Verdict

The authors compared their new "linked" model against the old "separate dice" model. The new model was a better predictor. It didn't just get the average scores right; it also got the relationship between the teams right. The old model failed to capture the fact that when one team scores, the other often scores less. By using this new method, the authors created a more realistic picture of how football matches actually play out, showing that the game is a dynamic conversation between two teams, not two separate monologues.

In short, the paper proves that football goals are connected, that the home crowd is a powerful force for the home team only, and that when the home team scores, the away team often finds it harder to score back. It's a reminder that in football, every goal changes the story for both sides.

Technical Summary: A Bayesian Bivariate Conditional Poisson Regression for Goal Dependence in the English Premier League

Problem Statement
Football match outcomes are characterized by a pair of non-negative integer counts: goals scored by the home team and goals scored by the away team. While traditional scoring models, such as the seminal work by Maher (1982) and subsequent extensions by Dixon and Coles (1997) and Karlis and Ntzoufras (2003), have established the importance of home advantage and team strengths, they often rely on the assumption that home and away goal counts are independent or, in the case of standard bivariate Poisson models, restrict dependence to non-negative correlation. This restriction is a significant limitation, as match dynamics often involve negative dependence; for instance, a team protecting a lead may suppress the opponent's scoring opportunities ("parking the bus"). Furthermore, existing models often struggle to simultaneously capture the joint dependence structure and the distinct, asymmetric effects of match-level covariates, such as stadium attendance and fouls, on the two scoring processes.

Methodology
To address these limitations, the authors propose a Bayesian Bivariate Conditional Poisson (BCP) regression model. The methodology is built upon the parameterization introduced by Berkhout and Plug (2004) and reparameterized by Piancastelli et al. (2023a).

Model Structure: The model defines the joint distribution of home goals ( $Y_1$ ) and away goals ( $Y_2$ ) by specifying one marginally and the other conditionally. Specifically, $Y_1 \sim \text{Poisson}(\lambda_1)$ and $Y_2 | Y_1 = y_1 \sim \text{Poisson}(\mu_2 e^{\phi y_1})$ , where $\mu_2$ is a function of $\lambda_1, \lambda_2,$ and the dependence parameter $\phi$ . This formulation allows for both positive and negative correlations, determined by the sign of $\phi$ .
Covariate Integration: The model incorporates match-specific covariates via log-linear specifications for the intensity parameters $\lambda_{1,i}$ $λ_{1, i}$ and $\lambda_{2,i}$ $λ_{2, i}$ . The covariates include:
- Log-transformed stadium attendance ( $\log(\text{Attendance} + 1)$ ) to handle zero-attendance matches (e.g., during the pandemic).
- Fouls committed by the home and away teams.
- Separate regression coefficients are estimated for home and away goals, allowing for asymmetric effects of these covariates.
Inference Framework: The study employs a full Bayesian inference framework.
- Priors: An empirical Bayes approach is used, where prior distributions for the regression coefficients and the dependence parameter are centered at zero with standard deviations derived from frequentist maximum likelihood estimates (MLEs).
- Computation: Posterior distributions are approximated using Hamiltonian Monte Carlo (HMC) sampling implemented in Stan. Four chains with 2,000 iterations each were run.
- Model Validation: Model fit is assessed via Posterior Predictive Checks (PPCs), comparing observed summary statistics (means, variances, correlations, and match outcomes) against replicated datasets generated from the posterior.
- Model Selection: Predictive performance is evaluated using Leave-One-Out Cross-Validation (LOO-CV) and the Expected Log-Predictive Density (ELPD). The authors compare two directional specifications (Home $\to$ Away vs. Away $\to$ Home) and an independent Poisson baseline.

Key Contributions

Flexible Dependence Structure: The paper introduces a BCP regression framework that explicitly accommodates negative correlation between home and away goals, overcoming the non-negative correlation constraint of standard bivariate Poisson models.
Asymmetric Covariate Effects: The model allows for separate estimation of how attendance and fouls influence home versus away scoring rates while simultaneously modeling the dependence between the two.
Tractable Parameterization: Unlike other recent conditional Poisson specifications (e.g., Petretta et al., 2025), this approach utilizes a parameterization where marginal moments and correlation metrics are analytically well-defined and available in closed form.
Empirical Application: The framework is applied to a dataset of 1,140 English Premier League (EPL) matches across three distinct periods: pre-pandemic (2018–19), restricted-attendance (2020–21), and post-pandemic (2023–24).

Results

Dependence: The posterior estimate for the dependence parameter $\phi$ in the preferred Home $\to$ Away specification is $-0.107$ (95% CI: $[-0.147, -0.066]$ ). This indicates a statistically significant negative correlation, suggesting that an increase in home goals is associated with a reduction in the expected number of away goals.
Model Performance: Both BCP specifications (Home $\to$ Away and Away $\to$ Home) outperformed the independent Poisson model in terms of joint predictive accuracy (ELPD-LOO). However, the Home $\to$ Away specification demonstrated superior conditional predictive accuracy, justifying its selection as the primary model.
Covariate Effects:
- Attendance: There is a positive and significant association between stadium attendance and home-team scoring (posterior mean $\approx 0.024$ ). Conversely, the association between attendance and away-team scoring is centered at zero with a credible interval including zero. This confirms the asymmetry of crowd influence.
- Fouls: The effects of fouls were less decisive. While away fouls showed a weak positive association with home goals, the 95% credible intervals for both foul coefficients included zero, indicating no strong evidence of a decisive foul effect at this level of analysis.
Posterior Predictive Checks: The BCP model successfully reproduced the observed negative correlation between goals and the proportion of home losses, whereas the independent Poisson model failed to capture the goal correlation.

Significance and Claims
The paper claims that the BCP regression provides a more realistic joint description of football match outcomes by capturing the negative dependence inherent in tactical interactions (e.g., lead protection). The results highlight that crowd presence primarily boosts home-team scoring without a corresponding effect on away-team scoring, reinforcing the concept of home advantage. The authors emphasize that while the model improves upon traditional independence assumptions and captures structural dependence, the estimated attendance effect should be interpreted as an adjusted association rather than a definitive causal effect, given the confounding factors of pandemic conditions and team composition across the selected seasons. The study concludes that the BCP framework offers a robust, interpretable tool for analyzing goal dependence and covariate impacts in football analytics.