If we have additional knowledge that the average roll of our die is 3, then we want to maximize -P(1)·Log(P(1)) - P(2)·Log(P(2)) - P(3)·Log(P(3)) - P(4)·Log(P(4)), given that the sum is 1 and the average is 3. We can either plug in the constraints and set partial derivatives to zero, or we can use a maximization technique like Lagrange multipliers.

I've never been able to understand this.

Surely the correct course of action in this situation is to have a prior for the possible biases of the die, say the uniform prior on {x in R^4 : x1+x2+x3+x4=1, xi>=0 for all i}, and then update Bayesianly by restricting to the subset where the average is 3. Then to find the distribution for the outcomes of the die we integrate over this.

I'm pretty sure this doesn't give the same distribution as maxent, and I can't think of a prior that would. (I think my suggested prior gives the "straight lines" distribution that you wanted!)

So when are each of these procedures appropriate? I agree that maxent is a good way to assign priors, but I think that when you have data you should use it by updating, rather than by remaking you prior.