
Find the ERM density
Suppose that $\displaystyle X_1,X_2,...,X_n $ are i.i.d. to $\displaystyle N( \mu , \sigma ^2 ) $ with both parameters unknown, we wish to estimate the density function of X with a family of densities as learners and the negative log loss function $\displaystyle L(f) =  \log (f) $
Find the ERM density.
So I know I have to find $\displaystyle \^ {R} (f_{ERM}) = argmin \frac {1}{n} \sum ^n_{k=1} (  \log f_{X_k} ) $
Now I'm stuck here since I don't know what to do, I thought about taking the derivative and set it equals to zero. But then I get $\displaystyle \sum ^n_{k=1} \frac {1}{f_{X_k}} = 0 $
But then how would you solve it? Thanks.