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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.

A $100(1-\alpha )\%$ credible set is a measurable subset $C(x)\subseteq (0,1)$ with posterior probability $1-\alpha$:In ordinary language, given the observed data and the chosen prior model, the probability that the parameter lies in this set is $1-\alpha$. Such a set need not be an interval or be unique; its defining requirement is posterior mass.