I was thinking about a problem of a colored balls in a box.
If we have 3 different possible ball colors distributed uniformly. The probability that the ball appearing in the box is any one of the colors would obviously be 1/3.
On the other hand, if we have 3 different possible ball colors distributed in an unknown way, there is no obvious way to know for certain what the probability of a color occurring is. But it still seems like the obvious "best guess" of that probability is 1/3.
A more extreme example is if we have a ball of an unknown color, with no extra information given. I don't know what possible colors can even exist, much less the chance of them occurring. I feel like the best guess of all possibilities would be 0 in this case, just like a continuous distribution.
I was trying to think about this in terms of Bayesian probability, but it seems like it's not very useful since you don't known the prior or the marginal. Maybe its just that there's not enough information to solve in a meaningful way. And so far the answers I saw from others on this type of problem is to basically "assume uniform distribution for everything". But then the usefulness of the answer from the uniform assumption becomes really "not useful".
Is there a mathematically correct "best guess"? Or some way to tell how good a guess is?