What is the significance of MLE = arithmetic mean?

Question

I am reading Jaynes' book on probability theory.

He explains that a particular derivation for the normal distribution was first recognized by Gauss, when he asked the following question. Say we have a sampling distribution $f_m(x)$ with location parameter $m$, so that $f_m(x)=f_0(x-m)$. Now assume we sample $X=\{x_0,...,x_n\}$ from this.

Now assume we place a restriction on $f$: the maximum likelihood estimate (MLE) of $m$ given $X$ must be equal to the sample mean $\bar x$.

The only probability distribution $f$ that satiafies this constraint is the normal distribution.

I find this first of all quite surprising, but more importantly, what is the significance of the requirement that MLE$(m)$=$\bar x$? Why should we pay any attention to the distribution that happens to satisfy that requirement?

Answer 1

It is an interesting requirement because it means that when you're looking for an estimator of the mean of the distribution, you just need to take the empirical mean and you get an excellent estimator (MLE, hence BLUE).

Two statements sound weird to me however:

• "The only probability distribution f that satisfies this constraint is the normal distribution."

I would think that the uniform distribution on [m-1,m+1] also satisfies it.

• "The normal distribution was first recognized by Gauss"

Although he didn't fully appreciate its importance, Laplace studied the normal distribution a few years before Gauss' birth.