= Credible interval
{title2=$\mathbb P(\Theta\in C(d)\mid D=d)=1-\alpha$}
{wiki}
A <credible interval> is an <interval> $C(d)$ containing a prescribed amount $1-\alpha$ of <posterior probability> for an unknown scalar <statistical parameter> $\Theta$. For a continuous <posterior distribution> with <posterior density> $p(\theta\mid d)$, its endpoints satisfy
$$
\mathbb P(a\leq\Theta\leq b\mid D=d)=\int_a^b p(\theta\mid d)\,d\theta=1-\alpha.
$$
The conditioning includes the chosen <statistical model> and <prior distribution>. For a discrete <posterior distribution>, exact equality may be impossible because of its <probability masses>; an interval can instead be required to contain at least the requested <posterior probability>. An interval compresses the full <posterior distribution> into a range whose uncertainty can be communicated in the parameter's own units. See https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/mit18_05_s22_statistics.pdf#page=68[MIT 18.05, Class 16: probability intervals].
For the eight-head, two-tail example in <Bayesian statistics>, the <posterior density> is $2860\theta^9(1-\theta)^3$ on $[0,1]$. Its <cumulative distribution function>, obtained by <integration> of this <polynomial>, is
$$
F(t)=286t^{10}-780t^{11}+715t^{12}-220t^{13},\qquad 0\leq t\leq1.
$$
Solving $F(a)=0.025$ and $F(b)=0.975$ by the <bisection method> gives
$$
a\approx0.461868461,\qquad b\approx0.909079605,\qquad F(b)-F(a)=0.95.
$$
These are numerical approximations to the exact <quantiles> defined by those equations; the endpoints were computed from the <cumulative distribution function>, rather than read off a plot. The resulting <equal-tailed credible interval> leaves 2.5% <posterior probability> on each side. Its shaded area is 95% of the total area under the <posterior density>.
\Image[/bayesian-credible-interval.png]
{title=Two different 95 percent credible intervals for the same Beta(10,4) posterior}
<Credible intervals> are not unique. The <one-sided credible interval> $[0,F^{-1}(0.95)]\approx[0,0.887334219]$ also has 95% <posterior probability>, but leaves all 5% outside it on the right. A <highest density region> instead selects points above a density threshold; it need not be an <equal-tailed credible interval>, and when the <posterior density> has separated peaks it need not be one <interval> at all. Thus a reported <credible interval> should specify its construction.
A 95% <credible interval> makes a statement about the <statistical parameter> conditional on these observed data and the Bayesian model. A 95% <confidence interval> instead refers to the <coverage probability> of a procedure under repeated sampling at a fixed parameter value. These are different conditioning statements: 95% <posterior probability> does not by itself imply 95% <coverage probability> for every fixed parameter. The <confidence interval> article develops that sampling interpretation.
Back to article page
