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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?

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If you are using equal-tailed credible intervals, these will inevitably fail to include the endpoints. Such intervals won't necessarily include the mode (though will always include the median). Instead, I think what you want is a 95% Highest Posterior Density Interval (95%HPDI). This takes the narrowest interval that includes 95% of the data and will, by definition, include the mode. However, it is possible that it won't include the median. Personally, I find HPDIs to generally be a more intuitive summary of a posterior distribution because they show you the range of values that are most likely according to the posterior.

Quite a nice description of highest density regions is available in this answer: https://stats.stackexchange.com/a/148475/53742

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