Let \Theta be the parameter space, let X_1,X_2,\dots ,X_ n be random variables, and let \alpha \in (0,1) be a fixed positive real number. Given a candidate Bayesian confidence regions, \mathcal{R}, we want to check whether it is indeed a confidence region within \alpha. That is, we want to check if
\mathbb {P}(\theta \in \mathcal{R}|X_1,X_2,\dots X_ n) \geq 1-\alpha ,
holds. Assuming that \Theta is a finite set, the probability above turns out to be
\mathbb {P}(\theta \in \mathcal{R}|X_1,X_2,\dots X_ n)=\sum _{\theta \in \mathcal{R}}P_1(\theta ),
where P_1(\cdot ) is some distribution supported on \Theta.
Which one of the probability distributions below gives the correct choice of P_1(\cdot )?
\pi (\theta ), the prior distribution on \theta.
L_ n(X_1,\dots ,X_ n|\theta ), the likelihood of the model.
\pi (\theta |X_1,X_2,\dots ,X_ n), the posterior distribution of \theta, conditional on X_1,X_2,\dots ,X_ n.
None of the above.