I'm reading about Fischer's Discriminant for multiple classes. In Duda and Hart's Pattern Recognition they write that
For the $c$-class problem, the natural generalization of Fisher’s linear discriminant involves $c − 1$ discriminant functions.
I'm not really sure why it's not $c$ discriminant functions. I know that if you have a score for $c-1$ classes, you can use something like softmax operation to get probability vectors for $c$ classes, as in $$ p(c_i) = \frac { e^{s_{c_i}}}{1+ \sum_{i=1}^{c-1} e^{s_{c_i}}}$$ for $i = 1, \ldots, c-1$ and $$p(c_i) = \frac{1}{1+ \sum_{i=1}^{c-1} e^{s_{c_i}}}$$ for $i=c$.
But I'm not sure how this extends to discriminant functions. Any insights appreciated.
