Thanks for the explanation of how it works for continuous random variables. I wonder about the maximizing the product of the sample probabilities. Is that maximizing for the samples in a specific given order? It seems for samples in any order it should be the product multiplied by n!. However n! is a constant is the minimization problem is the same.
Hi, not sure about your question but the order doesn’t matter
I have another question. In this section of Maximum Likelihood, if I understand correctly that the fundamental assumption of all the derivations, particularly writing the joint probability distribution in the product of individual marginal distribution requires the Independent and Identical distribution (i.i.d) .
But for me,
(1) at some point you only mentioned identical distribution, and some other points you only mentioned identical distribution. Was that did on purpose with restricted mathematical proof? Or should I understand that all what you said as i.i.d?
(2) What if the samples in X are NOT independent, but rather correlated? Then Eq.(18.7.10) cannot be written in the products of the individual probability. Then what will happen? And what modification will the method have to make in this section? What will changed in the conclusions? Are there good methods or references to deal with this problem?