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?
\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.
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.
The correct statement is: "None of the above."
Explanation: A Bayesian confidence region is constructed based on a prior distribution and posterior distribution of the parameter. It is not directly related to the confidence level like classical confidence intervals.
The statement "𝓡1 is necessarily a Bayesian confidence region for level 𝛼2" is incorrect because Bayesian confidence regions are not defined in terms of specific confidence levels.
The statement "𝓡1 is not necessarily a Bayesian confidence region for level 𝛼2 because it has insufficient probability mass" is also incorrect. The probability mass in a Bayesian confidence region is determined by the prior and posterior distributions, not by the specific confidence level.
Lastly, the statement "𝓡1 is not necessarily a Bayesian confidence region for level 𝛼2 because each level has a unique associated confidence region" is also incorrect. Bayesian confidence regions are not directly associated with unique confidence levels.
Therefore, none of the given statements are true.