A Poisson football goals model—from score rates to totals

Start with two expected goal rates

A simple football score model estimates one scoring rate for the home team and one for the away team. A Poisson distribution then assigns a probability to each possible goal count, such as zero, one, two or three.

Combining the two distributions creates a score matrix. Adding the cells with at least three total goals produces an Over 2.5 probability; adding the remaining cells produces Under 2.5.

Classical does not mean complete

Maher's 1982 work showed that independent Poisson models can describe football score frequencies reasonably well while still leaving systematic differences. Dixon and Coles later introduced adjustments for low-scoring outcomes and changing team strength.

Modern features can improve a baseline, but complexity is not automatically skill. Every additional feature increases the opportunity for leakage, overfitting and unstable behaviour outside the training sample.

Validate probabilities, not just picks

A model should be compared with a simple baseline and the market on data it did not train on. Calibration, Brier score, log loss and prospective closing-price comparison answer different questions and should remain separate.

• Train on information available at the historical cutoff.
• Calibrate on a separate period.
• Freeze the model before prospective evaluation.
• Report performance by league and probability band.
• Keep missing prices in the coverage denominator.