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  "body": "\n## Definition\n\nLet $\\theta$ be a real parameter, and let $\\Pi(\\cdot \\mid x)$ be its [[posterior distribution|0KJ7ZS8AJA]] given the observed data $x$.\nLet $\\alpha \\in (0, 1)$.\nA *credible interval* of level $1 - \\alpha$ is an interval $[a, b]$, with $a$ and $b$ depending only on $x$, such that\n$$\\Pi\\left([a, b] \\mid x\\right) \\ge 1 - \\alpha.$$\nThe *equal-tailed* credible interval has as endpoints the $\\alpha/2$ and the $1 - \\alpha/2$ quantiles of the posterior.\nSuppose the posterior has a unimodal density $\\pi(\\cdot \\mid x)$.\nThe *highest posterior density* interval is then the set $\\set{\\theta : \\pi(\\theta \\mid x) \\ge c}$, with $c$ chosen so that its posterior probability is $1 - \\alpha$.\n\n## Remarks\n\nExample: with a uniform prior on $p \\in (0, 1)$ and $n$ successes in $n$ Bernoulli trials, the posterior is $\\mathrm{Beta}(n + 1, 1)$, with distribution function $p^{n + 1}$ on $[0, 1]$.\nSo $[\\alpha^{1/(n+1)}, 1]$ has posterior probability $1 - \\alpha$, and it is a credible interval of level $1 - \\alpha$.\nNon-example: a frequentist [[confidence interval|0K4NMMJBG2]] is defined by its coverage over repeated samples, not by a posterior probability.\nFor the same data it can differ from every credible interval, and it need not have posterior probability $1 - \\alpha$.\nIn a parameter space of higher dimension the same notion is called a credible region or credible set.\n",
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