Let \mathcal{R_1} be a Bayesian confidence region of level \alpha _1 for a parameter heta given observations X_1,\dots ,X_ n. Let \alpha _2 be a another confidence level such that \alpha _2\geq \alpha _1. Which one of the statements below is true?

Question

Let \mathcal{R_1} be a Bayesian confidence region of level \alpha _1 for a parameter 	heta given observations X_1,\dots ,X_ n. Let \alpha _2 be a another confidence level such that \alpha _2\geq \alpha _1. Which one of the statements below is true?

\mathcal{R_1} is necessarily a Bayesian confidence region for level \alpha _2.

\mathcal{R_1} is not necessarily a Bayesian confidence region for level \alpha _2, because it has insufficient probability mass.

\mathcal{R_1} is not necessarily a Bayesian confidence region for level \alpha _2, because each levels has a unique associated confidence regions.

None of the above.

GPT 3.5 · Answer

None of the above is necessarily true. The correct statement is that \mathcal{R_1} is not necessarily a Bayesian confidence region for level \alpha _2, because each level has a unique associated confidence region. This means that different levels of confidence will have different regions. Just because \mathcal{R_1} is a confidence region for level \alpha _1 does not mean it will be a confidence region for level \alpha _2, especially if \alpha _2 > \alpha _1. The probability mass of \mathcal{R_1} may not be sufficient to satisfy the higher level of confidence.

Let \mathcal{R_1} be a Bayesian confidence region of level \alpha _1 for a parameter \theta given observations X_1,\dots ,X_ n. Let \alpha _2 be a another confidence level such that \alpha _2\geq \alpha _1. Which one of the statements below is true?