## 3 Methodological Procedure
### Regression Algorithms
Bayesian Ridge [31]: The implementation involves a linear regression model that incorporates probability distributions instead of relying on fixed point estimates, thus offering distributions as outputs, unlike the traditional approach that produces a single predicted value. Consequently, the independent variable, \(s\), is derived not as a single point but rather from a normal distribution, \(\mathbb{N}(\mu,\sigma)\), defined by its mean, \(\mu\), and variance, \(\sigma\). In this model, the variance is calculated as the square of the standard deviation, multiplied by the identity matrix, \(I\). The mean is represented by \(\beta_{j}^{T}X\), where \(\beta_{j}^{T}\) denotes the transpose of the coefficient vector \(\beta_{j}\), and \(X\) denotes the predictor variable. In Bayesian Ridge, both the likelihood and the prior play specific and complementary roles in defining the model and computing the posterior distribution of the parameters. The likelihood encapsulates the probability of observing the data given the parameters of the model. In Bayesian Ridge, the likelihood often characterizes how the input features, \(X\), relate to the target, \(y\), with some inherent noise. For a linear regression model, the relation is expressed as \[y=X\beta+\epsilon,\] (11) where \(y\) represents the observed data, \(X\) is the design matrix, \(\beta\) is the vector of regression coefficients, and \(\epsilon\sim\mathcal{N}(0,\sigma^{2})\) accounts for Gaussian noise. The variance of the noise is estimated as part of the Bayesian inference process. The estimation involves treating the variance (or its inverse, called precision, \(\alpha=1/\sigma^{2}\)) as a random variable with its own prior distribution, which is updated using the observed data to compute a posterior distribution. The likelihood function is given by \[p(y|X,\beta,\sigma^{2})\propto\exp\left(-\frac{1}{2\sigma^{2}}(y-X\beta)^{T}( y-X\beta)\right).\] (12)  
The prior defines initial assumptions about the parameters before examining the data. Different prior selections embody different levels of previous knowledge or hypotheses. Gaussian priors for the coefficients \(\beta\) are frequently applied in linear regression models:  
\[\beta\sim\mathcal{N}(\mu_{0},\Sigma_{0}), \tag{13}\]  
where \(\mu_{0}\) represents the mean vector (initial belief about the coefficients) and \(\Sigma_{0}\) is the covariance matrix, indicating the uncertainty surrounding this initial belief. 5. Decision Tree [39]: This supervised learning algorithm applies to regression and classification tasks using a hierarchical structure of nodes. A complete Decision Tree includes a root node, branches, internal nodes, and leaf nodes [40]. At every node in the algorithm, a feature along with a threshold is chosen to divide the data, with the purpose of optimizing a criterion that evaluates the \"purity\" or \"homogeneity\" of the resulting subsets. The data set undergoes successive splitting into smaller subsets until certain stopping conditions are fulfilled (e.g., reaching maximum depth or a minimum number of samples per node). For predicting a new sample's value or class, the algorithm navigates the tree according to the sample's feature values until it arrives at a leaf node. To determine the optimal split, the algorithm employs metrics such as Gini Impurity (refer to Equation (14)) or Entropy. \[G=1-\sum_{i=1}^{C}p_{i}^{2},\] (14) where \(p_{i}\) represents the fraction of samples belonging to class \((i)\) at a node, and \(C\) is the number of classes. 6. Deep Neural Networks [41]: A neural network (NN) is a subset of machine learning that is loosely inspired by the functioning of the human brain. The fundamental computational units in a neural network are called neurons, and these neurons are organized into layers. The size, number, and arrangement of layers can vary depending on the specific application [42].