DDE-1/Regression.md

@@ -95,7 +95,18 @@ 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. 
 
-...
+Let's look at a simple, 2D case. In the linear model, we know have only two trainable parameters:
+
+```math
+y_p(x, w) = w_0 + w_1x_1 
+```
+Herein, we will represent the the probabilities for the weights in terms of Gaussian distributions. If we withdraw weights from these prior information (assumed), what we end up with is alternative linear models:
+
+
+
+At this stage, the model is not trained at all. What happens as we see examples? Lets show the model the very first example (blue circle below). In the Bayesian learning, we use the likelihood function, p(y|x,w) -- probability of getting the true y value given the wights and the example. With the example we pass, the calculated likelihood is shown below (left). The true weights are shown as "+" for comparison. Here, the calculated likelihood estimates that the weights of the model should be around this zone. By using the likelihood and our prior, we now calculate the posterior probabilities, given in the middle:
+
+<img src="uploads/47f290dd30648b9cf47e1c597fc45a11/br2.png"  width="500">
 
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