## 2 Bayesian Belief Network and Quantitative Measures for Causal Inference
### Lyapunov-Based Divergence Measure for Causal Factor Identification  
Causal candidates, \(X_{i}^{\text{u}}\in\{\text{X}_{\text{i}}\backslash\text{e}\}\), are unobservable risk variables whose probabilistic state values diverge when the effect variable state transitions from state \"no\" to state \"yes\", given that the transition is mediated by the influence of the observable variables, e, i.e., transition is induced by the evidence variable. This definition follows the form of the definition of a maritime accident, which states that an accident is preceded by a certain combination of influential factors. In other words, when evidence variables are instantiated at a nominal state of a model, where the nominal state is one with no accident, then the effect, i.e., accident, variable, is instantiated to the state \"yes\" so as to mimic the dynamics leading to an accident, and it can be expected to observe sequential changes in the probabilities of unobservable factors \(\text{X}_{\text{i}}^{\text{u}}\). The sequential state values \(\text{P}_{\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{\text{u}}\,|\,\text{Y}= \text{${}^{\prime}$no}\text{${}^{\prime}$})\to\text{P}_{\text{i}}^{\text{u}}( \text{X}_{\text{i}}^{\text{u}}\,|\,\text{Y}=\text{${}^{\prime}$no}\text{${}^{ \prime}$};\text{e})\to\text{P}_{\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{ \text{u}}\,|\,\text{Y}=\text{${}^{\prime}$yes}\text{${}^{\prime}$};\text{e})\), where symbol \"\(\to\)\" denotes transition, provide novel information. By examining the rate of change of these values, a diverging behavior of unobservable causal factor can be identified. For example, the values:  
\[\delta\text{F}_{1}=\text{P}_{\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{ \text{u}}\,|\,\text{Y}=\text{${}^{\prime}$no}\text{${}^{\prime}$};\text{e})- \text{P}_{\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{\text{u}}\,|\,\text{Y}= \text{${}^{\prime}$no}\text{${}^{\prime}$})\]  
\[\delta\text{F}_{2}=\text{P}_{\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{\text{u} }\,|\,\text{Y}=\text{${}^{\prime}$yes}\text{${}^{\prime}$};\text{e})-\text{P}_ {\text{i}}^{\text{u}}(\text{X}_{\text{i}}^{\text{u}}\,|\,\text{Y}=\text{${}^{ \prime}$no}\text{${}^{\prime}$};\text{e})\]  
that are both positive, (\(\delta\text{F}_{1}>0\) &&\(\delta\text{F}_{2}>0\)), indicate the sequential growth of the probability of the causal factor \(\text{X}_{\text{i}}\), which is interpreted as a rise of the negative influence of the causal factor \(\text{X}_{\text{i}}\). The interpretation follows from the monotonicity property of BBN. The conceptual representation of possible parameter probability changes of \(\text{X}_{\text{i}}\) in two sequential steps is visualized in Figure 2 and it is the interest of the approach to identify only those with sequential decay of probability. The described process can be formalized with a dynamical state-space model [26; 27]:  
\[ \begin{split}\frac{\mathrm{d}\mathrm{p}_{\mathrm{i}}}{\mathrm{d}x}& =\lambda\mathrm{p}_{\mathrm{i}}(x)\\ \mathrm{p}_{\mathrm{i}0}&=\mathrm{p}_{\mathrm{i}}(x_ {0})\end{split}  \tag{4}\]  
Function \(\mathrm{p}(x)\) is a path of evolution of the homogenous dynamical system with respect to change in the variable \(x\), where \(\mathrm{p}_{0}\) is an initial value. The solution of the model is:  
\[\mathrm{p}_{\mathrm{i}}(x)=\mathrm{p}_{\mathrm{i}0}\mathrm{e}^{\lambda_{ \mathrm{i}}(x-x_{0})} \tag{5}\]  
The component \(\mathrm{e}^{\lambda_{\mathrm{i}}(x-x_{0})}\) is a state transition matrix. Constant \(\lambda_{\mathrm{i}}\) is known as the Lyapunov exponent [28; 29], which describes how small distances between nearby points in the state space change with time in continuous systems or change with the sample instance in discrete systems, and serves to identify stable diverging of chaotic behavior of the system. There are as many Lyapunov exponents as there are sample instances. The average constant \(\lambda\) has a positive value when the system is unstable, negative when the system is stable, and a zero value indicates marginal stability.