Alright, I expect this is a silly question, but I don't actually know, so.
Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean $\mu$ and its variance $\sigma^2$. Maximum Entropy would have me say that I should choose the Normal distribution as the one that best fit my knowledge about it.
However, suppose I have an utility function about this variable that goes $U(x)=x^4$. Then, my expected utility for drawing from that variable's distribution is $EU(x) = \int\limits_{-\infty}^{+\infty}\!\!x^4\ p(x)\ \mathrm{d}x$. If I choose to use MaxEnt, then that's just $3\sigma^4$. However, if the "true" distribution actually followed by that random variable is, say, the Student's t, then my Expected Utility would diverge to infinity.
If I treat the distribution as if I don't know what it is, then I'd have that $p(x)=\int\limits_{D\in\mathcal D}\!\!p(x|D)\ p(D)\ \mathrm{d}D$ where $\mathcal D$ is the space of all possible distributions. In that case, my expected utility is $EU(x) = \int\limits_{-\infty}^{+\infty}\!\int\limits_{D\in\mathcal D}\!\!x^4\ p(x|D)\ p(D)\ \mathrm{d}D\ \mathrm{d}x$ which I expect is not the same as the one I get by using the Normal except for very few priors over $\mathcal D$.
My questions then are: how do I deal with that? What prior distribution over $\mathcal D$ gives me the same result as MaxEnt would? In what sense is the Normal my best guess for my state of ignorance? How do I guarantee that this doesn't diverge? How does having the Normal as the "representative" of my subjective uncertainty take other possible distributions such as the Student's t into account?