I'm trying to apply the maximum entropy principle to estimate the probability of observing a rare event. For example, if I rolled a *fair* die (6 sided), I could estimate that the probability of rolling any number 1-6 is $\displaystyle \frac{1}{6}$, and the expected outcome as $\displaystyle \bar{x} = \sum\limits_{i=1}^{6} x \frac{1}{6} = 3.5$. However, if I observed something rare, say $\displaystyle 2.5 < \bar{x} < 2.7$, I might not be able to make the assumption that the die is fair.

I'd like to use Gibbs, the large deviations principle, or the maximum entropy principle (via gradient descent) to determine the probability distribution of each possible die outcome (i.e. $\displaystyle p(x=1), p(x=2), \dots p(x=6)$ given the fact that I observed this rare event. There are many examples online of how to apply these principles, but none that I have been able to find that explain how to take a rare event into account.

If anyone can point me to such a worked example, or provide one here, I would be immensely grateful!