Bayesian networks are a commonly used method of representing conditional probability relationships between a set of variables in the form of a directed acyclic graph (DAG). Determination of the DAG which best explains observed data is an NP-hard problem [1]. This problem can be stated as a constrained optimisation problem using Integer Linear Programming (ILP). This paper explores how the performance of ILP-based Bayesian network learning can be improved through ILP techniques and in particular through the addition of non-essential, implied constraints. There are exponentially many such constraints that can be added to the problem. This paper explores how these constraints may best be generated and added as needed. The results show that using these constraints in the best discovered configuration can lead to a significant improvement in performance and show significant improvement in speed using a state-of-the-art Bayesian network structure learner.

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The increasing awareness of climate change and human capital issues is shifting companies towards aspects other than traditional financial earnings. In particular, the changing behaviors towards sustainability issues of the global community and the availability of environmental, social and governance (ESG) indicators are attracting investors to socially responsible investment decisions. Furthermore, whereas the strategic importance of ESG metrics has been particularly studied for private enterprises, little attention have received public companies. To address this gap, the present work has three aims—1. To predict the accuracy of main financial indicators such as the expected Return of Equity (ROE) and Return of Assets (ROA) of public enterprises in Europe based on ESG indicators and other economic metrics; 2. To identify whether ESG initiatives affect the financial performance of public European enterprises; and 3. To discuss how ESG factors, based on the findings of aims #1 and #2, can contribute to the advancements of the current debate on Corporate Social Responsibility (CSR) policies and practices in public enterprises in Europe. To fulfil the above aims, we use a combined approach of machine learning (ML) techniques and inferential (i.e., ordered logistic regression) model. The former predicts the accuracy of ROE and ROA on several ESG and other economic metrics and fulfils aim #1. The latter is used to test whether any causal relationships between ESG investment decisions and ROA and ROE exist and, whether these relationships exist, to assess their magnitude. The inferential analysis fulfils aim #2. Main findings suggest that ML accurately predicts ROA and ROE and indicate, through the ordered logistic regression model, the existence of a positive relationship between ESG practices and the financial indicators. In addition, the existing relationship appears more evident when companies invest in environmental innovation, employment productivity and diversity and equal opportunity policies. As a result, to fulfil aim #3 useful policy insights are advised on these issues to strengthen CSR strategies and sustainable development practices in European public enterprises.