I'm studying EM algorithm from this lecture notes. Here I understand the generalized form of EM algorithm, but what confuses me is derivation of EM (Expectation Maximization) algorithm for mixture of Gaussian model. On page 7 of the mentioned notes, following derivation is given:
Till step 2, it is all clear, but in step 3, how summation sign ($\sum_j$) is removed while taking derivative w.r.t $u_l$? Also why we are taking derivative w.r.t $u_l$ instead of $u_j$? Any help would be appreciated.
The derivation is trivial. "j" indicates an index between 1 and k just for the sum operation and is not related to the index "l" of your interest. To take the derivative w.r.t. $u_l$, we can focus on only the $l$-th term of the summation.