A Bayesian network is a graphical probabilistic belief network that represents the conditional dependencies among uncertain variables, which can be both objective and subjective. We present a Bayesian network model for forecasting Association Football matches in which the subjective variables represent the factors that are important for prediction but which historical data fails to capture. The model (pi-football) was used to generate forecasts about the outcomes of the English Premier League (EPL) matches during season 2010/11 (but is easily extended to any football league). Forecasts were published online at www.pi-football.com prior to the start of each match. In this paper, we demonstrate that a) using an appropriate measure of forecast accuracy, the subjective information improved the model such that posterior forecasts were on par with bookmakers' performance; b) using a standard profitability measure with discrepancy levels at ≥ 5%, the model generates profit under maximum, mean, and common bookmakers' odds, even allowing for the bookmakers' built-in profit margin.Hence, compared with other published football forecast models, pi-football not only appears to be exceptionally accurate, but it can also be used to 'beat the bookies'.

Objectives1) To develop a rigorous and repeatable method for building effective Bayesian network (BN) models for medical decision support from complex, unstructured and incomplete patient questionnaires and interviews that inevitably contain examples of repetitive, redundant and contradictory responses; 2) To exploit expert knowledge in the BN development since further data acquisition is usually not possible; 3) To ensure the BN model can be used for interventional analysis; 4) To demonstrate why using data alone to learn the model structure and parameters is often unsatisfactory even when extensive data is available.MethodThe method is based on applying a range of recent BN developments targeted at helping experts build BNs given limited data. While most of the components of the method are based on established work, its novelty is that it provides a rigorous consolidated and generalised framework that addresses the whole life-cycle of BN model development. The method is based on two original and recent validated BN models in forensic psychiatry, known as DSVM-MSS and DSVM-P.ResultsWhen employed with the same datasets, the DSVM-MSS demonstrated competitive to superior predictive performance (AUC scores 0.708 and 0.797) against the state-of-the-art (AUC scores ranging from 0.527 to 0.705), and the DSVM-P demonstrated superior predictive performance (cross-validated AUC score of 0.78) against the state-of-the-art (AUC scores ranging from 0.665 to 0.717). More importantly, the resulting models go beyond improving predictive accuracy and into usefulness for risk management purposes through intervention, and enhanced decision support in terms of answering complex clinical questions that are based on unobserved evidence.ConclusionsThis development process is applicable to any application domain which involves large-scale decision analysis based on such complex information, rather than based on data with hard facts, and in conjunction with the incorporation of expert knowledge for decision support via intervention. The novelty extends to challenging the decision scientists to reason about building models based on what information is really required for inference, rather than based on what data is available and hence, forces decision scientists to use available data in a much smarter way.

Successful implementation of major projects requires careful management of uncertainty and risk. Yet such uncertainty is rarely effectively calculated when analysing project costs and benefits. This paper presents a Bayesian Network (BN) modelling framework to calculate the costs, benefits, and return on investment of a project over a specified time period, allowing for changing circumstances and trade-offs. The framework uses hybrid and dynamic BNs containing both discrete and continuous variables over multiple time stages. The BN framework calculates costs and benefits based on multiple causal factors including the effects of individual risk factors, budget deficits, and time value discounting, taking account of the parameter uncertainty of all continuous variables. The framework can serve as the basis for various project management assessments and is illustrated using a case study of an agricultural development project

When developing a causal probabilistic model, i.e. a Bayesian network (BN), it is common to incorporate expert knowledge of factors that are important for decision analysis but where historical data are unavailable or difficult to obtain. This paper focuses on the problem whereby the distribution of some continuous variable in a BN is known from data, but where we wish to explicitly model the impact of some additional expert variable (for which there is expert judgment but no data). Because the statistical outcomes are already influenced by the causes an expert might identify as variables missing from the dataset, the incentive here is to add the expert factor to the model in such a way that the distribution of the data variable is preserved when the expert factor remains unobserved. We provide a method for eliciting expert judgment that ensures the expected values of a data variable are preserved under all the known conditions. We show that it is generally neither possible, nor realistic, to preserve the variance of the data variable, but we provide a method towards determining the accuracy of expertise in terms of the extent to which the variability of the revised empirical distribution is minimised. We also describe how to incorporate the assessment of extremely rare or previously unobserved events.