normalized exponential function
\[p(\mathcal{C}_k \vert x) = \frac{\exp(a_k)}{\Sigma_j \exp(a_j)}\]
where \[a_k = \ln p(\mathbf{x} \vert \mathcal{C}_k) p(\mathcal{C}_k)\]
the softmax function terminology derives from the fact that if \[a_k > a_j\] for all \[j \neq k\] then \[p(\mathcal{C}_k) \approx 1\] and \[p(\mathcal{C}_j) \approx 0\]