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 -1.
No, we cannot determine the value of 𝔼(𝜃∈𝐼|𝑋_1,…,𝑋_𝑛) based solely on the information that the probability is greater than or equal to 𝜖, where 𝜖>0.
To determine 𝔼(𝜃∈𝐼|𝑋_1,…,𝑋_𝑛), we need to know the underlying distribution of 𝑋_1,…,𝑋_𝑛 and the true value of 𝜃. The given information only tells us that the probability is greater than or equal to 𝜖, but it doesn't provide any additional information about the distribution or the true value of 𝜃.
Therefore, we cannot determine 𝔼(𝜃∈𝐼|𝑋_1,…,𝑋_𝑛) based solely on the given information. The answer is -1.
No, we cannot determine the value of 𝑃(𝜃∈𝐼|𝑋₁,𝑋₂,…,𝑋ₙ) using only the information that 𝑃(𝜃∈𝐼|𝑋₁,𝑋₂,…,𝑋ₙ) ≥ 𝜖 for some 𝜖 > 0.
To evaluate 𝑃(𝜃∈𝐼|𝑋₁,𝑋₂,…,𝑋ₙ), we need more information about the distribution of 𝑋₁, 𝑋₂, …, 𝑋ₙ and the parameter 𝜃. The interval 𝐼 is constructed based on the observed values, but without knowing the specific distribution and parameters, we cannot determine the exact probability 𝑃(𝜃∈𝐼|𝑋₁,𝑋₂,…,𝑋ₙ).