Suppose that the parameter space is \Theta = \{ \theta _1,\dots ,\theta _ n\}, and assume that one of these, \theta _ i, is the true parameter \theta. (Here we are using the frequentist set-up– the parameter is not modeled as a random variable.)
Let X_1,X_2,\dots ,X_ n be observations. You construct an interval
I=\left[\overline{X}_ n - \sqrt{X_1^2+\cdots +X_ n^2},\overline{X}_ n+\sqrt{X_1^2+\cdots +X_ n^2}\right].
Assume that you have observed X_1,\dots ,X_ n, and you are interested in
\mathbb {P}(\theta \in I|X_1,\dots ,X_ n).
Suppose that you have access to an all-knowing genie, who says that the probability above is greater than or equal to \epsilon, for some \epsilon >0. Using only this information, can you determine
\mathbb {P}(\theta \in I|X_1,\dots ,X_ n)?
If yes, enter your answer to the input box below. If not, enter -
No, we cannot determine the value of P(θ∈I|X₁,...,Xₙ) based solely on the information that the probability is greater than or equal to ε. We would need more information about the distribution of the observations and the true parameter in order to calculate the probability.
No, we cannot determine the exact value of $\mathbb{P}(\theta \in I|X_1,\dots ,X_n)$ using only the information that the probability is greater than or equal to $\epsilon$.
The reason is that the probability depends on the actual distribution of the observations $X_1, X_2, \dots, X_n$ and the true parameter $\theta$. Without any further information about the distribution or the specific values of $X_1, X_2, \dots, X_n$, it is not possible to calculate the probability.
We would need additional information, such as the distributional assumptions or the specific values of $X_1, X_2, \dots, X_n$, in order to determine the value of $\mathbb{P}(\theta \in I|X_1,\dots ,X_n)$.