The definition of the Q function includes these probabilities. We use the analytic for of Q to get the expressions that maximize it (by equating the derivative to 0). (12.9) is such an expression. It uses the posterior probabilities we calculated in the E-step.

]]>On page 4 the EM algorithm for GMM is being described.

It is not clear what exactly happens with Q(teta,teta-t) - it seems we do not derive it when we evaluate the parameters at the M-step, but rather equation 12.7 which appears later. Is equation 12.7 equals to Q(teta,teta-t)? if it is - why? if not - then what exactly do we derive?