Why Gauss distribution is a necessary condition for the equality of maximum likelihood estimate and sample mean

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).

• The proof in Jaynes is rather awfully documented. You can find a more interesting [reference here], in particular check out "The Proof" on page 105-106. In particular, the heart of the "proof" relies on the assumption that the maximum likelihood equation holds for all samples, in particular the adhoc choice that Jaynes gives. : med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/… – Alex R. Jan 1 '16 at 6:09
• @AlexR. : Thanks. But I think you've give the wrong link, it is the link of Jaynes' proof, which is in my question. – Bin Wang Jan 1 '16 at 8:30
• Whoops. Here's the right link: maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/… – Alex R. Jan 1 '16 at 20:41