I am trying to think of a good motivation for maximum likelihood estimation.
Given a set of random variables $X_1, \ldots, X_n \sim f_X(x_1, \ldots, x_n |\theta)$, the maxmum likelhood estimation problem finds $\theta$ that maximizes $f_X(x_1, \ldots, x_n |\theta)$ given $x_1, \ldots x_n$.
But is a good physical analogy as to why we want to do this?
I was thinking of...you have a class room of students, you pick 10 of them and get their average height, and you assume that average height is the average height for the entire class. However, there is no maximizing in my analogy. So it doesn't work.