I'm reading Probability Theory: The Logic of Science by Jaynes. And while I'm reading section 7.3, he proofed that Gauss distribution is a necessary condition for the equality of maximum likelihood estimate and sample mean.
But he just derived Gauss distribution from a special case of sample mean, how does that imply a necessary condition? It is like if we know
a + b = 0,then we assume
a = 1, b = -1, from this special case we get
a - b = 2, but that doesn't implies
a - b = 2 is a necessary condition of
a + b = 0.
( You may read Chapter 7 of this book online if you don't have this book).