Credible interval

Builds on Posterior probability Bayesian inference

Definition

Let $\theta$ be a real parameter, and let $\Pi(\cdot \mid x)$ be its posterior distribution given the observed data $x$ . Let $\alpha \in (0, 1)$ . A credible interval of level $1 - \alpha$ is an interval $[a, b]$ , with $a$ and $b$ depending only on $x$ , such that $\Pi\left([a, b] \mid x\right) \ge 1 - \alpha.$ The equal-tailed credible interval has as endpoints the $\alpha/2$ and the $1 - \alpha/2$ quantiles of the posterior. Suppose the posterior has a unimodal density $\pi(\cdot \mid x)$ . The 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 $1 - \alpha$ .

Remarks

Example: 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 $[0, 1]$ . So $[\alpha^{1/(n+1)}, 1]$ has posterior probability $1 - \alpha$ , and it is a credible interval of level $1 - \alpha$ . Non-example: a frequentist confidence interval is defined by its coverage over repeated samples, not by a posterior probability. For the same data it can differ from every credible interval, and it need not have posterior probability $1 - \alpha$ . In a parameter space of higher dimension the same notion is called a credible region or credible set.

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Construction of the reals from Cauchy sequences concept
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The rationals form a field statement
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The rationals form an ordered field statement
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Prior probability concept
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Posterior probability concept
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Bayesian inference concept
Credible interval definition

Tag	`K7Y7AMMPK0`
URI	`m2://K7Y7AMMPK0`
Kind	definition
Writing	W0 stub
MSC	62F15, 62F25
Defines	Credible interval
Uses	Posterior probability, Bayesian inference

Credible interval

Definition

Remarks

Reading path

Further reading

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