Credible set ($\mathbb{P}(\theta \in C\mid x)=1-\alpha$)

A credible set is a measurable set of parameter values containing specified posterior probability. At credibility level $1-\alpha$, it satisfies ${\int }_C\pi (\theta \mid x)d\theta =1-\alpha$ for a continuous posterior. A credible interval is a credible set that is an interval. Equal-tailed and highest-density choices add selection rules; the posterior-mass requirement alone does not make the set unique. Its probability statement is conditional on the observed data and prior model, whereas a frequentist confidence interval is interpreted through repeated-sample coverage.