## 2 Material and Methods  
### Bayesian Regularization  
Bayesian networks serve as statistical tools utilized to model causal connections and uncertainties among distinct events. These networks depict data relationships within a probabilistic framework, visually outlining the interrelations among a sequence of events and employing probability distributions to articulate the dependence of each event on others [42].  
For example, consider the assessment of a health condition where symptoms exhibited by a patient may correlate with specific illnesses. Bayesian networks are well-suited for modeling such scenarios, as they can ascertain the likelihood that symptoms like a cough, fever, and headache indicate a particular ailment [43; 44].  
These networks facilitate the computation of event probabilities and provide insights into their interrelationships. In essence, Bayesian networks aid in understanding how an event unfolding within a given context may impact other scenarios. As a result, they find extensive applications across diverse domains, including decision making, prognostication, and risk assessment.  
A Bayesian network is a graphical representation of an uncertain collection of quantities, also known as a directed acyclic graph (DAG), along with a set of conditional probability tables that correspond to the structure of the DAG [45]. The network consists of a sequence of probability nodes (ovals) and directed arcs connecting these nodes. Nodes represent stochastic variables, defined as a set of discrete states, with each state associated with a probability measure. The arcs between variables denote conditional probability dependencies, signifying the likelihood of a dependent variable Y (child node) being in a specific state for each combination of states of the preceding variables X (parent node). The presence of a directed arc from node X to node Y indicates that X has a direct influence on Y, specified by the conditional probability P(Y | X). Bayesian networks do not allow for directed loops. Despite its concise design, the graphical representation offers a comprehensive probabilistic depiction of the scenario. A key feature of Bayesian networks is their ability to infer from evidence observed at any node, with new knowledge traversing the network and updating all variables in the model under [PERSON] rule [46].  
Bayesian regularization, as a network training function, optimizes weight and bias values using Levenberg-Marquardt optimization. This method minimizes a combination of squared errors and weights, identifying the optimal combination to construct a network with robust generalization. Implemented in MATLAB R2013b as the \"trainbr\" trainer, the training process depends on the function's parameters. Validation vectors are employed to halt training prematurely if the network's performance on the validation vectors does not improve or remains constant for \"max_fail\" epochs consecutively. Bayesian regularization does not use a separate validation set but includes it in the training set [10]. Test vectors serve as an additional check for the network's generalization but do not impact the training. Enabling validation by setting \"max_fail\" to any strictly positive number ensures weight/bias minimization with shorter training times. Regarding the performance function, this approach shares the same constraint as \"trainlm\".