This understanding of validation is more appropriate for systems and models where it is unfeasible to compare the output of a risk model with observations or experiments. In this Section, the overall framework for the construction of the Bayesian network is introduced. First, some basic issues concerning Bayesian networks are briefly outlined, showing how BNs can accommodate the adopted risk perspective.

In mathematical terms, Bayesian networks (BNs) represent a class of probabilistic graphical models, defined as a pair Δ = G(X, A), P ( Koller and Friedman, 2009 and Pearl, 1988), where G(X, A) is the graphical component and P the probabilistic component of the model. G(X, A) is in the form of a directed

acyclic graph (DAG), where the nodes (X) represent the variables X = X1, …, Xn in the considered problem Trametinib cost and the arcs (A) represent the probabilistic conditional (in)dependence relationships between the variables. P consists of a set of conditional probability tables (CPTs) P(Xi|Pa(Xi)) for each variable Xi, i = 1, …, n in the problem. Pa(Xi) signifies the set of parents of Xi in G: Pa(Xi) = Y ∊ X. Thus: P = P(Xi. A Bayesian network encodes a factorization of the joint probability distribution (JDP) over all variables in X: equation(5) P(X)=∏i=1nP(Xi|Pa(Xi))From Eq. (5), it follows that BNs have desirable properties Pifithrin-�� molecular weight for describing uncertainty about oil spills in ship–ship collisions, conditional to impact scenarios.

In particular, when an assessor expresses his uncertainty about the impact scenarios using a set of parent nodes, this uncertainty can be propagated through the model to attain an expression of uncertainty about the possible oil spill sizes. To achieve a full assessment of uncertainty and bias in line with the risk perspective of Eq. (4), a qualitative description of Methisazone U and B supplements the BN. As illustrated in Fig. 2, the BN is constructed from an integration of two main elements: a submodel GI linking the damage extent to ship particulars and oil outflow and a submodel GII linking the impact scenarios to the damage extent. First, the resulting oil outflow for product tankers is determined from outflow calculations in a range of damage scenarios using a set of representative product tankers. For these tankers, limited data is available concerning cargo tank number and configuration. The more detailed tank arrangement needed for oil outflow calculations is estimated based on a model presented by Smailys and Česnauskis (2006). The data obtained from subsequent oil outflow calculations is applied in a Bayesian learning algorithm to construct the first submodel of the BN. This submodel GI(XI, AI) consists of nodes and arcs related to the ship particulars, damage extent and oil outflow. Its construction is elaborated in Section 4.