Credible intervals

An interval containing a stated posterior probability of the parameter; unlike a confidence interval it can be read directly as a probability statement about the parameter, given the model and prior.

\left[\,Q\!\left(\tfrac{1-L}{2}\right), \; Q\!\left(\tfrac{1+L}{2}\right)\right], \qquad P(\theta \in [l, u] \given \text{data}) = L

Equal-tailed credible interval (engine default)

$Q$: quantile function of the posterior distribution
$L$: credible level, e.g. 0.95

\text{HPD: shortest } [l, u] \text{ with } F(u) - F(l) = L, \text{ where } f(l) = f(u)

Highest posterior density interval (unimodal posterior)

$f, F$: posterior density and cdf

When to use

Summarising a posterior distribution (e.g. after a beta–binomial update).
Equal-tailed: equal posterior probability $(1-L)/2$ beyond each end; invariant to transformations. HPD: the shortest interval; every point inside is more plausible than any point outside; useful for skewed posteriors.

Assumptions

The likelihood and prior are reasonable; the interval inherits both.

How to interpret

"Given the model, the prior and the data, there is a 95% probability that θ lies between l and u."
Contrast: a 95% confidence interval is produced by a procedure that captures the fixed true value in 95% of repeated samples; a single computed confidence interval either contains it or not, and the 95% describes the method.
With weak priors and large samples the two intervals are often numerically close, but their meanings differ.

Common mistakes

Reporting a credible interval without stating the prior.
Calling a confidence interval a credible interval (or vice versa) because the numbers match.
Asking for an HPD interval from a U-shaped or flat posterior, where it is not a single interval (the engine refuses these).

Example

Posterior Beta(13, 29) (12 successes in 40 trials, uniform prior): 95% equal-tailed interval 0.181 to 0.455; 95% HPD interval 0.175 to 0.449 (slightly shorter and shifted towards the mode, because the posterior is right-skewed).