Given the first two moments, the maximum entropy distribution over $\mathbb{R}$ is known the be the normal distribution. What is the analogue for a distribution over $[0,1]$ given either only the expectation or with the variance as well?
Is there a multivariate extension, analogue to the multivariate normal distribution?
The solution will be a normal distribution truncated to the interval $[0,1]$. The details are messy, and in practice some numerical work will be needed. The proof follows the proof in the unrestricted case, the differences occurs first when we have to find the Lagrange multipliers. But note that the $\mu,\sigma^2$ parameters of the normal not need to coincide with the same parameters from the restrictions. For there to be a solution we need that the restrictions satisfy $0\le\mu\le 1,\quad 0<\sigma^2\le \mu(1-\mu)$.
For the multivariate "box" case, exactly the same can be said. The restrictions on the parameters will be messy.
