i stumbled upon the following formula while studying the concept of exponential loss. I try to understand this formula. I read it as follows: The function that minimizes the expected exponential loss (i.e. the arg min (f(x) ) is equal to 1/2 log Pr(Y=1 |x)/Pr(Y=-1|x). I have however trouble understanding on how to visualize f(x), and how the division on the right leads to a function f(x)? any intuition much appreciated!
$$ f^*(x) = \arg\min_\limits{f(x)} E_{Y|x} \left( e^{-Yf(x)} \right) = \frac{1}{2} \log \frac{P(Y=1|x)}{P(Y=-1|x)} $$
For fixed $x$, call $p = P(Y=1|x)$. Then $1-p = P(Y=-1|x)$.
Let $G(f) = E_{Y|x}\left( \exp(-Yf) \right) = p\exp(-f) + (1-p)\exp(f)$.
Setting $\frac{dG}{df} = 0$ yields the result.