An Underrated Prior Distribution for Proportions. The Logistic–Normal for Dynamical Football Predictions

Abstract

The result of a football match in terms of Home–Win, Draw or Away–Win can be modelled by considering the observed outcome as a realization of a Multinomial random variable with three mutually exclusive events over a single trial. Most applications consider the Dirichlet distribution to represent the prior uncertainty about the Multinomial’s proportion parameters, mainly because of conjugacy and the reduced number of parameters. As alternative we propose to use the Logistic–Normal, a multivariate prior distribution for proportions but to which little attention has been paid. This approach was motivated by the question – Are women’s and men’s football leagues equally predictable? The models developed are applied to the main Portuguese women’s and men’s football leagues over seven seasons, starting from 2016–2017 up to 2022–2023. The work also provides estimates of latent team-specific strengths and addresses the variability between and within seasons, along with insights of each team’s home advantage.