Solution
= Solution
Given the observed data, a $100(1-\alpha)\%$ <credible interval> $[a,b]$ for the scalar parameter satisfies
$$
\boxed{\mathbb P(a\leq\theta\leq b\mid\boldsymbol x)=1-\alpha}.
$$
The probability refers to the <posterior distribution>, proportional to $\pi(\theta)\prod_i f(x_i\mid\theta)$ on the parameter space. It conditions on the observations; it is not the repeated-sampling <coverage probability> definition of a <confidence interval>.