Wilson interval with zero trials

Question

I'm calculating Wilson score interval for some number of trials, so I have a question:
is it correct to assume this interval as [0 , 1] when you have zero trials?

Answer 1

What is the value of $\hat p = X/n$ when there are no trials? I don't see how the formula for the Wilson interval can be applied if there is no data.

In the Wikipedia link the word "equivalently," between the first and second formulas for the interval, tacitly assumes $n$ is a positive integer. However, if you use the second form anyway for $n = 0$ then you do get, formally, $1/2 \pm 1/2$ or $(0,1)$ for a CI of any desired level of confidence. (If you're playing out this fantasy to the end, you could say $(0,1)$ is an 'all purpose CI' for $p$ which is assumed from the start to lie in $(0,1)$.)

In a Bayesian framework, one might choose the 'flat' prior $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1).$ With no data, the posterior could be taken to be the same as the prior and one Bayesian 95% probability interval would be $(.025,.975).$