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.