## 2 Bayesian Belief Network and Quantitative Measures for Causal Inference  
A Bayesian belief network (BBN) is an acyclic graph-based model consisting of nodes and probabilistic relationships among nodes that show the influences [2]. In a qualitative sense, nodes consist of N random variables V = {V\({}_{1}\),V\({}_{2}\),...,V\({}_{\text{N}}\)} and a set of directed edges E among nodes represent direct influence, i.e., a directed link from node V\({}_{i}\) to V\({}_{j}\) is causally interpreted as \"V\({}_{i}\) causes V\({}_{j}\)\" [2]. In terms of quantitative formulation, BBN is the joint probability distribution of values v = {v\({}_{1}\),v\({}_{2}\),...,v\({}_{\text{N}}\)} of N random variables V = {V\({}_{1}\),V\({}_{2}\),...,V\({}_{\text{N}}\)}:  
\[\text{P}(\text{v}_{1},\text{v}_{2},\ldots,\text{v}_{\text{n}})=\prod_{i}\text {P}(\text{v}_{i}|\text{Pa}(\text{v}_{i})) \tag{1}\]  
where Pa(v\({}_{i}\)) are the parents of variable V\({}_{i}\), and P(v\({}_{i}\)\({}_{1}\)\({}_{1}\)) is a local conditional probability associated with variable V\({}_{i}\)[19], which describes a parent-child relation. Figure 1: The BBN framework for maritime accident risk factor assessment. To fully construct the BBN, the first requirement is to define the problem domain through identification of a relevant set of variables \(\mathrm{V}\) constituting the problem being modelled. Second, relations among the variables are made to establish the graphical structure of the model, while in the next step, variable states and their relevant initial probabilities are assigned. Different information sources are used for variable identification, graph construction, and probability value assignment, such as a literature review, experts' beliefs, repository data, and empirical data. The BBN construction can be automated in domains with an abundance of data through structure learning and parameter optimization; however, the adoption of the BBN framework for modelling in maritime accidents, where data is sparse or not available, is tedious work that requires the involvement of an interdisciplinary team for maritime domain problem analysis, model structuring, and parametrizing, as will be described in Section 3. Once developed, the BBN model represents a formal model for inference and causal reasoning [20], and can be exploited for predictive and diagnostic analysis to extract the explanations of the domain through belief update via forward and backward message passing [1]. In order to interpret BBN model outputs through both predictive analysis with cause-to-effect queries, and diagnostic analysis with combined cause-to-effect and effect-to-cause queries, the variables of a BBN are related to causal reasoning system notation. A causal set \(\mathrm{X}=\{\mathrm{X}_{1}\),\(\mathrm{X}_{2}\),...,\(\mathrm{X}_{n}\}\) of n = 1,... \(\mathrm{r}\); \(\mathrm{r}=\mathrm{N}-1\), random variables \(\mathrm{X}\subset\mathrm{V}\) of BBN is a set of risk factors that can directly or indirectly cause an accident. The remaining variable from the set \(\mathrm{V}\backslash\mathrm{X}\) is a targeted event, i.e., maritime accident, denoted as Y. Each causal variable \(\mathrm{X}_{i}\in\mathrm{X}\) can further be related in a cause-effect manner to variables from \(\mathrm{X}\backslash\mathrm{X}_{i}\), as indicated by the link direction. In regard to the modular definition of the Bayesian network [21], causes at different levels of the network can be treated as effects, which is exploited when performing both predictive and diagnostic analysis of hypothetic scenarios and real world data. Through this modularity mechanism, a BBN accident model allows for arbitrary cause-effect and effect-cause queries of type \"What's the probability of grounding given traffic distribution is high?\", or \"What's the probability of being off course given traffic distribution is high, there's no radar and personal condition is poor?\" to be answered.