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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 satisfyThe 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 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, isSolving $F(a)=0.025$ and $F(b)=0.975$ by the bisection method givesThese 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.

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.

An equal-tailed credible interval uses quantiles of a continuous posterior distribution to leave posterior probability $\alpha /2$ in each tail. It is also called a central credible interval; central refers to equal tail probabilities, not equal distances from the posterior mean. For the beta distribution $\mathrm{Beta}(10,4)$ in credible interval, the 95% endpoints are approximately $0.461868461$ and $0.909079605$, whose midpoint $0.685474033$ differs from the posterior mean $5/7$.

Let $F_{A,B}$ denote the distribution function of the posterior beta distribution, with $A=a+s$ and $B=b+n-s$. This distribution is continuous with positive density on $(0,1)$. Its equal-tailed credible interval isCompute the two quantiles by solving ${\int }_0^{{\theta }_L}\pi (\theta \mid x)d\theta =\alpha /2$ and ${\int }_{{\theta }_U}^1\pi (\theta \mid x)d\theta =\alpha /2$. Each excluded tail has mass $\alpha /2$, leaving posterior mass $1-\alpha$ between the endpoints.

An equal-tailed credible interval allocates equal posterior probability to its two excluded tails:For a continuous strictly increasing posterior cumulative distribution function $F$, the endpoints are $F^{-1}(\alpha /2)$ and $F^{-1}(1-\alpha /2)$.

A highest posterior density interval instead includes the parameter values of greatest posterior density: there is a cutoff $c$ such that the retained region is a superlevel set of $\pi (\theta \mid x)$, of posterior mass $1-\alpha$. Every interior retained value has density at least as large as every excluded value, apart from possible ties. For a unimodal continuous density this is an interval; when both endpoints lie inside the support they have equal posterior density. For a multimodal posterior distribution, a highest-density region may be a union of intervals, rather than one interval. Equal tail probabilities and highest density are different criteria.

Let $F$ be the cumulative distribution function of a gamma distribution with shape $A=a+{\sum }_i{x}_i$ and rate $B=b+n$; explicitly,The posterior distribution is continuous and has positive density on $(0,\infty )$, so its quantiles give the requested equal-tailed credible interval:Equivalently, $2B\theta \mid x\sim {\chi }_{2A}^2$, so the endpoints can be written as the corresponding chi-squared distribution quantiles divided by $2B$. Either form excludes posterior probability $\alpha /2$ from each tail.

No. A permitted counterexample is $a=1$ with all observed counts zero. Then $A=1$ and the posterior distribution is an exponential distribution with rate $B=b+n$, whose density decreases strictly from the left boundary. Its equal-tailed credible interval isHere $L>0$. This excludes values just above zero whose posterior density is larger than that of retained values near $U$, violating the highest-density criterion. The actual highest posterior density interval starts at the support boundary and solves $1-e^{-Bh}=1-\alpha$:Writing zero as the endpoint causes no problem because a single point has zero posterior probability. Thus even within the gamma family the two intervals need not agree. More generally, for an interior unimodal gamma distribution, highest-density endpoints have equal densities, whereas equal-tailed endpoints need not.

Put $M_0=\alpha +x_0\beta$. As a linear combination of independent normal random variables under the posterior distribution,The shortest 95% credible interval isHere $\Phi$ is the standard normal distribution cumulative distribution function. To justify shortest length, consider all intervals of a fixed length $d$. If $f$ is this normal distribution posterior density, the interval probability ${\int }_a^{a+d}f(t)dt$ has derivative $f(a+d)-f(a)$ with respect to $a$. The normal distribution is symmetric and its density strictly decreases with distance from its mean, so this probability is uniquely maximized when the midpoint is $m_0$. The centered interval can therefore attain 95% posterior probability with the least length. It is both an equal-tailed credible interval and a highest posterior density interval.

The target is the experiment's mean at $x_0$. A posterior predictive distribution for a new noisy observation would instead have variance $v_0+{\sigma }^2$; that extra observation variance does not belong in the requested interval.

An equal-tailed credible interval for a continuous posterior distribution has probability $\alpha /2$ below its lower endpoint and $\alpha /2$ above its upper endpoint. If $F$ is the posterior cumulative distribution function, the endpoints are $F^{-1}(\alpha /2)$ and $F^{-1}(1-\alpha /2)$.

A highest posterior density interval is a credible interval consisting of points of highest posterior density: every included point has density at least as high as every excluded point, apart from null-set and boundary choices. It is a posterior highest density region when that region is an interval. For a continuous unimodal density with two interior endpoints, their densities agree. A mode at a support boundary can instead force one endpoint to that boundary; a multimodal posterior can have a disconnected highest-density region rather than one interval. Equal tails and highest density need not give the same interval.