Link between Maximum Likelihood and Maximum Probability How can I see that the maximum likelihood approach finds the parameter values of the probability distribution that maximize the probability of the observed sample?
Maximum likelihood is not the maximum of probability in general because for, say, two continuous random variables $X_1, X_2$ we have $P(X_1=x_1,X_2=x_2)=0$.
 A: It does not find a probability distribution. It finds some values for parameters of a fixed, apriori assumed as being the true, probability. So, if the true probability is the probability you assumed, then it fits parameter values of that probability function for which the data at hand achives it's maximum likelihood.
In the expression "it maximizes the probability of the observed sample" there are two components. The distribution family (the functional form of the distribution) which is asumed by you when you find parameters. There is no proof for that. You work with it or not. And the fitted estimated values for probability parameters. That estimates for parameters together with the functional form is the probability you for which you optimize the observed samples.
[Later] I think the confusion comes from the usage of probability word. Sometimes it means a probability distribution family and sometimes it means a probaility distribution with known parameter values. I think the later meaning is more appropriate, keeping separate a term for distribution family, but I confess I don't keep rigor, probably from a bad habit.
