# A Bayesian In-Play Prediction Model for Association Football Outcomes
## 6 Conclusions and Future Work  
In this paper, on the basis of Bayesian method, we propose a dynamic strength model which relaxes assumption of the constant teams' strengths and permits applying in-match performance information to calibrate teams' strengths. We test our model against the birth process model where teams' strengths are assumed to be constant during the match based upon betting performance on three common markets. With the intention of comprehensively verifying the performance for a match, we calculated the average return by considering betting every 5 min. The results demonstrate that our model can achieve a positive return and significantly outperform the birth process model on the over-under market. This extends to the outcome betting market when the threshold exceeds 0.06--in this circumstance our model succeeds in obtaining positive returns and therefore outperforms the birth process model (which also cannot achieve positive returns under different threshold values). However, applied to the point-spread market, our model does not clearly improve the forecasting of the score difference. However, this defect should be considered against the fact that our forecasting method has a clear and advantageous application to over-under market and Win/Draw/Loss betting market.  
Although we have presented some promising results for our dynamic calibration strengths model with in-match information, we believe that further improvements can be envisaged and put into effect. First, it is important to acknowledge that the model failed to improve prediction accuracy for goal difference--this suggests that future engagements should attempt to develop a model focused upon goal difference. Second, the present model only uses in-match goal times information; diversification beyond this limited point to engage with separate event information such as red cards could contribute clear research benefits. One possible approach is to introduce covariates into the baseline strength parameters \(\alpha\) and \(\beta\), i.e., using covariates to describe the team strength. Furthermore, the important covariates can be selected by using Bayesian model choice. The relevant literature can be referred to in [31; 32].  
Conceptualization, Q.Z. and J.S.; methodology, Q.Z., K.S. and J.S.; software, Q.Z.; validation, Q.Z., K.S. and J.S.; formal analysis, Q.Z.; investigation, Q.Z.; resources, Q.Z.; data curation, Q.Z. and J.S.; writing-original draft preparation, Q.Z.; writing-review and editing, K.S. and J.S.; visualization, Q.Z.; supervision, J.S.; project administration, J.S.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Major Project grant number 2017 ZX06002006.  
We would like to thank the guest editors and reviewers for their valuable comments and suggestions which helped us to improve this paper. We would also thank Beijing StatusWin Lottery Operations Technology Ltd for providing the data.  
Figure 6: Average return from betting on outcome market for various values of the threshold \(\tau\).