Most certainly. I did not intend to question the usefulness of maxent models. I just find myself a bit uneasy with maxent as a derivation of and/or explanation of the ubiquity of the normal distribution, given the two issues mentioned above when we're talking about the continuous case. I was wondering if you might have some insight into the issue which could remedy this feeling of uneasiness :)

And regarding the moments, it's just that the normal distribution is the only distribution with a finite number of non-zero moments. Therefore, constraining higher order moments is not so straight forward.

Also might be worth noting that technically, if one was sufficiently UNreasonable, one could constrain the target distribution to take on specific values given specific inputs. This would not be very useful in any real-world applications. Then choosing between constraining only the first moment ("minimal" constraints), or constraining each point you've observed to take on the normalized frequency ("maximal" constraints) becomes entirely up to you. Therefore I don't see quite how maxent models give us the tools for deciding between complexity and accuracy, as the maxent models can be on either end of the spectrum depending on what constraints we choose.

(Unsure if you were implying that it did, but nonetheless it might be something to note.)