# Integrating Multi-Omics Using Bayesian Ridge Regression with Iterative Similarity Bagging
## 3 Materials and Methods
### Bayesian Ridge Regression  
Bayesian Ridge Regression (BRR) was employed as a feature selection strategy. This approach involves calculating coefficients to estimate the relevance score of each feature. Bayesian regression focuses on utilizing the Bayesian approach, wherein the estimation of linear models is conducted by considering probability distributions rather than singular point values [25]. The posterior probability of model parameters is influenced by training inputs and outputs. Additionally, the ridge method is integrated into Bayesian regression to mitigate the issues of model complexity and multicollinearity through coefficient shrinkage. Most omics data are typically characterized as small-scale, making Bayesian analysis appropriate for such scenarios. The process combines the existing information about the parameter, known as the prior parameter distribution, with the observed data [27]. Bayesian Ridge Regression (BRR) differs from traditional Ridge Regression by utilizing a probabilistic framework and including prior distributions for the regression coefficients [65]. Traditional Ridge Regression aims to minimize the sum of squared errors while imposing a penalty term that restricts the regression coefficients' L\({}^{2}\) norm (Euclidean norm) [65]. Including this penalty term alleviates the problem of multicollinearity and enhances the stability of the regression coefficient estimation, particularly in the context of high-dimensional data. In contrast, BRR treats regression coefficients as stochastic variables and assigns a prior distribution to them, typically Gaussian [27]. This Bayesian approach facilitates the integration of prior knowledge or beliefs about coefficient distributions into the modeling process. During parameter estimation, Bayesian Ridge Regression (BRR) not only maximizes data likelihood but also updates with observed data. This results in a posterior distribution representing the regression coefficients' uncertainty. The final approximations are typically derived by computing posterior mean or mode, striking a balance between prior information and observed data [65; 66; 67]. The introduction of uninformative priors on hyperparameters allows regularization akin to L\({}_{2}\) regularization in Ridge Regression and classification. This regularization seeks maximum a posteriori estimation under a Gaussian prior over the coefficients w with precision \(\lambda^{-1}\). Instead of setting lambda manually, it is possible to treat it as a random variable to be estimated from the data [66]. The output \(y\) is assumed to be Gaussian-distributed around \(Xw\) to create a fully probabilistic model [68]. \[p(y|X,w,\alpha)=\mathcal{N}(y|Xw,\alpha) \tag{5}\]  
where \(\alpha\) is once more considered a random variable that needs to be estimated from the data. As previously described, Bayesian Ridge Regression estimates a probabilistic model for the regression problem, employing a spherical Gaussian as the prior for the coefficient \(w\)[65]. \[p(w|\lambda)=\mathcal{N}\Big{(}w\big{|}0,\lambda^{-1}I_{p}\Big{)} \tag{6}\]  
The priors over \(\alpha\) and \(\lambda\) are selected as gamma distributions, which are conjugated prior for Gaussian precision. The resulting model, similar to the classical Ridge Regression, is known as Bayesian Ridge Regression. During model fitting, the parameters \(w\), \(\alpha\), and \(\lambda\) are estimated jointly, with the regularization parameters \(\alpha\) and \(\lambda\) estimated by maximizing the log marginal likelihood [65; 69]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Type** & **Raw Data** & **Processed** \\ \hline Drugs & 98 & 98 \\ Cell lines & 1124 & 555 \\ Gene expression & 11,833 & 11,712 \\ Copy number alteration & 24,960 & 24,959 \\ Single-nucleotide mutation & 70 & 54 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The number of samples and features of genomics data in the used GDSC dataset. This study utilized BRR to select informative features based on drug response.