DDE-1/Regression.md

@@ -85,12 +85,20 @@ If you are more curious about its derivation and the underlying mathematics of t
 
 In the “regression problem” (see lecture notes for “ode to learning” and “regression”), we have discussed that selection of model complexity is needed to be compatible with the data (dimensions, volume) in order to minimize the over-fitting problem. We have also seen that we can force regularization on the lost function to give additional penalty for the over-fitting. Nonetheless, its impact is limited as the nature of the base function (i.e. our scientific hypothesis) is still there, affecting the overall behaviour of the ML model deployed. 
 
-As an alternative path, we can follow the probabilistic learning to alleviate the over-fitting in the regression analysis.
-  
+As an alternative path, we can follow the probabilistic learning to alleviate the over-fitting in the regression analysis. Let us consider a simple, linear regression case:
+
+```math
+y_p(x, w) = w_0 + w_1x_1 + . . . + w_ix_i
+```
+
+In the probabilistic world, the trainable parameters (w) will be represented by probability distributions, represented by Gaussians. These initial statements act like prior knowledge on the fitting. In the next step, we update our PDFs via Bayesian update, obtaining the posterior distributions --again as Gaussians (we love Gaussians as it is very easy to work with them: combining Gaussians will yield a new Gaussian distribution). By examining  new cases (training data), we can build up a better understanding of the system behaviour iteratively in this probabilistic landscape. 
+
 ...
 
 ## Additional Sources
 
+Explore the additional resources for more!   
+
 - [Seeing theory: linear regression](https://seeing-theory.brown.edu/#secondPage/chapter6)
 - [Cross validation](https://scikit-learn.org/stable/modules/cross_validation.html?highlight=repeatedkfold)
 - [Basics of regression](https://ml-cheatsheet.readthedocs.io/en/latest/linear_regression.html)