# A Bayesian In-Play Prediction Model for Association Football Outcomes
## 1 Introduction  
Association football is the most popular sport in the world and betting on outcomes has a long tradition. In addition, football constitutes the fastest growing gambling market [1]. As a result, the modelling and prediction of football match outcomes has become increasingly popular [2; 3; 4]. The prediction of an individual match outcome is a challenging task, which can be broken down into two types: prediction pre-match and prediction in-play. Pregame prediction previously used the Negative Binomial model for association football scores over the Poisson model Reep [5]. However, since the seminal paper by [PERSON] [6], the Poisson distribution goal-based data analysis, in which match results are generated by the two teams' attack and defence parameters, has been widely deployed [7; 2; 8; 9]. The plausibility of Poisson regression models for the interpretation and prediction of association football (soccer) scores were examined in [10]. In addition, because the Poisson count model only applies in instances where the data are time-homogeneous and equi-dispersed, [PERSON] [11] proposed the count model based on Weibull interarrival times, which were capable of managing both under and over-dispersed data. [PERSON] et al. [12] applied the Weibull model for forecasting association football scores, and the results demonstrated it performed better than the Poisson count model. Recently, [PERSON]. [13] introduced a probabilistic model for forecasting whether the teams in a soccer competition will reach the targetthat they set at the beginning of the championship. [PERSON]. [14] developed a suitable probability model for studying the points achieved by a team in a football match. A substantial amount of literature has focused upon improving the aforementioned models, for example, [PERSON]. [15] developed a hierarchical Bayesian Poisson model in which the scoring rates of the teams are convex combinations of parameters estimated from historical data and the additional source of the betting odds. [PERSON] et al. [16] made use of the specialists' opinions and the FIFA ratings as prior information, and applied the Bayesian methodology for predicting match outcomes. [PERSON] et al. [1] presented a Bayesian network model for predicting football outcomes with subjective variables which represented the factors that were important for prediction but historical data failed to capture; [PERSON] [17] proposed Bayesian forecasting models to allow the strengths of each team to vary over time. [PERSON]. [18] accounted for team strengths using the line-up of fielded players, a mixed effects model used to identify a player's true goal scoring ability. [PERSON]. [19] presented a player-based model for football scores which takes into account the abilities of the players on each team. In addition, their basic model for the scoreline in a football match is a bivariate Weibull count model proposed by [12]. Live betting is very popular within betting markets and prediction in-play is therefore a significant factor that deserves closer consideration. However, among the huge amount of literature published on prediction of association football games, only a few papers concentrated on the in-play prediction. [PERSON] et al. [20] developed a pure birth process model, where the processes of goal times of home and away teams are taken to be two nonhomogeneous Poisson processes. In attempting to align themselves with practical circumstances, [PERSON] et al. [21] proposed a discrete-time and finite-state Markov chain model that is grounded within the Poisson processes, where no more than one goal in a minute had happened, except during time intervals (44, 45) and (89, 90) in consideration of injury time, and a recursive algorithm was derived to accurately calculate the outcome probability. [PERSON] [22] and [PERSON] [23] studied the effects of other events such as cards in match.