Bayesian Approach to Interval Estimation

In the Bayesian approach to interval approximation for the unknown parameter(s), $\theta$, of a model is based on the posterior distribution $\pi (\theta \mid \underset{―}{x})$.

The interval of $(1-\alpha )$ of credibility is any interval $(L,U)$ that satisfies $$\int_L^U \pi(\theta\mid \underline x) , d\theta = 1-\alpha$$
the intervals of probability are not unique. We can choose, for example, an interval where the tails do this $$\int_{-\infty}^L \pi(\theta\mid \underline x), d\theta =\int_U^\infty \pi(\theta\mid \underline x), d\theta = \alpha/2 $$
or a unilateral where $L=-\mathrm{\infty }$ or $U=\mathrm{\infty }$. in the cases where the posterior distribution is unimodal, it is possible to adopt a high posterior density (HPD), where $\pi (L\mid \underset{―}{x})=\pi (U\mid \underset{―}{x})$. In this case this the minimum length interval.