# Logistic regression cost function intuition

My question is regarding the LR cost function from andrews ML course (http://feature-space.com/en/document50.pdf , page -5)

$cost= \frac{1}{m}[ -y \times \log(\psi) - (1-y) \times \log(\kappa) ]$

The vector y holds values for the digits (1-10), so if we plug these values in the cost function then the cost function takes ambiguous values. For instance if y=5, then the cost function will have both the parameters

$cost (y=4) = \frac{1}{m} [ -4 \times \log(\psi) -(1+4) \times \log(\kappa) ]$

As per Andrew's lecture I remember him saying that only one of the log terms would remain inside the cost function as if classified correct $\log(\psi)$ term remains else $\log(\kappa)$ remains.

$y$ always takes on values of 1 or 0, as you noted. For the multi-class problem, you're going to solve for the "one vs. all" case. You'll need to transform your $y$ vector into a vector of 1's and 0's depending on the class you are minimizing for. So for the number 5, you'll solve for $P(y=5)$ vs. $P(y \ne 5)$. You repeat that for all your digits. You come up with 10 different $h_{i}{\theta}$'s, i.e. $h_{1i}{\theta}$, $h_{2i}{\theta}$, ...