I am estimating an unknown variable which is between 0 and 1. I start off with a uniform prior on [0, 1]. Then I update it given some evidence (flip the coin a few times). Now I have a posterior.
The posterior pdf is positive everywhere. So, when I take a 95% credible interval, it never includes the endpoints 0 and 1. In this case, the true value might be on these endpoints, so I don't want to rule them out.
What could I do about this? Is it solvable with a continuous pdf? Or does my prior need mass points on 0 and 1? Or, is there another way to solve this?