## 2 The Knowledge Engineering Problem  
### Dynamic Bayesian Networks  
A dynamic Bayesian network (DBN) is an extension of a Bayesian network (BN) to model probabilistically time-series [32; 33]. More specifically, it can be interpreted as a sequence of time slices. Each time slice (\(t\)) encodes a static BN.  
A BN is a Directed Acyclic Graph (DAG) that enables reasoning under uncertainty within the Bayesian framework [34]. This graphical structure is defined through a set of random variables that are represented through nodes, and their conditional dependencies that are encoded through arcs. When an arc between the nodes \(X\) and \(Y\) is specified as \(X\to Y\), then the node \(X\) is defined as the parent node of \(Y\).  
Let us define a set of random variables \(X=\{X_{1},\ldots,X_{n}\}\) with \(n\) total number of BN nodes. We can order the nodes such that the parents of a node \(X_{i}\), also known as parent configuration, are a set  
Figure 1: The knowledge acquisition (KA) approach discussed in this paper.  
of variables \(pa(X_{i})=\{X_{q},\ldots,X_{i-1}\}\subseteq\{X_{1},\ldots,X_{i-1}\}\). In BNs, a node is conditionally independent of its non-parent ancestors, therefore  
\[p(X_{i}|X_{1},\ldots,X_{i-1})=p(X_{i}|pa(X_{i})). \tag{1}\]  
As described in [33], in a DBN, we can denote with \(U=\{U_{1},\ldots,U_{T}\}\) a set of random variables, which can be partitioned in \(U_{t}=(X_{t},Z_{t},Y_{t})\), where \(t=1\ldots T\) represents the time slice, \(X_{t}\) represents the observed variables (i.e., input variables), \(Z_{t}\) represents hidden (latent) variables, and \(Y_{t}\) the output variables of a state-space model. It has to be highlighted how in general the term _dynamic_ refers to the dynamic nature of the system modelled, but the network is not changing over time.  
A DBN can be defined as a pair, \((B_{1},B\rightarrow)\), where \(B_{1}\) is a BN defining the prior \(p(U_{1})\), and \(B\rightarrow\) is a _two-slice_ temporal Bayes net (2 TBN), which encodes the conditional probability \(p(U_{t}|U_{t-1})\) through a DAG. More specifically, this relation can be expressed as follows:  
\[p(U_{t}|U_{t-1})=\prod_{i=1}^{n}p(U_{t}^{i}|pa(U_{t}^{i})). \tag{2}\]  
For notional reasons, we can assume that a first-order Markov assumption holds. Therefore, the parents of a node, \(pa(U_{t}^{i})\), can be in the same time slice or in the previous one. More in general, arcs can be defined across more than two slices.  
By _unrolling_ the 2 TBN over \(T\), we obtain the _semantics_ of the DBN. More specifically, we can define the joint probability distribution as:  
\[p(U_{1:T})=\prod_{t=1}^{T}\prod_{i=1}^{n}p(U_{t}^{i}|pa(U_{t}^{i})). \tag{3}\]  
In the engineering community, more and more DBNs are adopted as uncertainty representation and inference tools. One of the reasons for that can be found in the good balance between expressiveness and tractability [33].  
The BN knowledge engineering task includes the definition of the relevant variables, their relations, and the conditional probability distributions [1]. The first two aspects determine the BN structure. The last one, which determines the associated BN conditional probability tables (CPTs), consists of specifying for each \(X_{i}\) the expression in Equation (1). The CPTs are often elicited from experts that might be requested to answer several questions such as: _\"What is the probability of a fish transshipment taking place given that the two ships of interest are performing a rendezvous?\"_.