Past exam of the statistics course of the University of Oxford / 2014 / 3 / a

Given observed data $x$, a credible interval $[L(x),U(x)]$ has posterior probability $1-\alpha$:For a continuous posterior distribution with probability density function $\pi (\theta \mid x)$, this is ${\int }_{L(x)}^{U(x)}\pi (\theta \mid x)d\theta =1-\alpha$. The probability is conditional on the observed data and concerns the statistical parameter under its posterior distribution; it does not assert frequentist coverage probability.

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.