add some certainty stuff

src/implementation.tex

@@ -15,9 +15,15 @@ The model with the weights of the current loop iteration predicts pseudo predict
 \end{equation}
 
 Those predictions might have any numerical value and have to be squeezed into a proper distribution which sums up to 1.
-The Softmax function has exactly this effect: $\sum^\mathcal{S}_{i=1}\sigma(z)_i=1$
+The Softmax function has exactly this effect: $\sum^\mathcal{S}_{i=1}\sigma(z)_i=1$.
+Since we have a two class problem the Softmax results in two result values, the two probabilities of how certain one class is a match.
+We want to calculate the distance to the class center and the more far away a prediction is from the center the more certain it is.
+Vice versa, the more centered the predictions are the more uncertain the prediction is.
+Labels $0$ and $1$ result in a class center of $\frac{0+1}{2}=\frac{1}{2}$.
+That means taking the absolute value of the prediction minus the class center results in the certainty of the sample~\eqref{eq:certainty}.
 
 \begin{align}
+    \label{eq:certainty}
     S(z) = | 0.5 - \sigma(\mathbf{z})_0|  \; \textit{or}  \; \arg\max_j \sigma(\mathbf{z})
 \end{align}