How does an equal-tailed credible interval differ from a highest posterior density interval?

## Two rules for slicing the same posterior A credible interval is any range holding a stated share of the posterior probability. Since infinitely many ranges hold 95% of a continuous posterior, you need a rule. Two are standard. **Equal-tailed interval.** Take the value below which 2.5% of the posterior lies, and the value above which 2.5% lies. The interval between them is the 2.5th to 97.5th posterior percentile range. Equal probability is discarded on each side, hence the name. **Highest posterior density (HPD) region.** Take the shortest set holding 95% of the posterior. Equivalently, pick a horizontal cut through the posterior density and include every value whose density sits above the cut, lowering the cut until the included mass reaches 95%. Every value inside has density at least as high as every value outside - which is exactly the property the name describes. ## Where they agree and where they part If the posterior is symmetric and unimodal - a Gaussian posterior, or a Beta posterior with roughly balanced counts - the two rules give almost the same endpoints, and the choice does not matter for reporting. Skew is where they separate. Consider two conversions out of a hundred trials with a uniform prior on the rate. The posterior is Beta(3, 99): its mean is about 2.9%, its mode is 2%, and it has a long right tail. The equal-tailed 95% interval runs roughly 0.61% to 6.96%, a width of about 6.35 percentage points. The HPD interval runs roughly 0.31% to 6.21%, a width of about 5.90 percentage points. The HPD is shorter and its whole span slides toward the mode, because the density near the lower end is higher than the density in the far right tail that the equal-tailed rule insists on keeping. The general pattern: on a right-skewed posterior the equal-tailed rule buys symmetry of *probability* at the cost of carrying a stretch of thin right tail while excluding denser values on the left. ## Boundaries Suppose you observe zero events in a hundred trials with a uniform prior. The posterior is Beta(1, 101), a density that is highest at zero and decays from there. The HPD region is `[0, 2.9%]` - it runs right up to the boundary, because that is where the density is highest. The equal-tailed interval cannot do that: by construction it must leave 2.5% of the mass below its lower endpoint, so it starts at a small but strictly positive value, roughly 0.03%, and runs to about 3.6%. The equal-tailed rule therefore excludes zero purely as an artefact of the rule, not because the data disfavour zero. ## Multiple modes If the posterior has two separated peaks with a low-density valley between them, the HPD region at 95% can come out as two disjoint chunks - one around each peak - because the valley has lower density than the tails of the peaks. The equal-tailed interval always returns one contiguous range, so it bridges the valley and reports values the posterior considers relatively implausible. Neither is wrong; they answer different questions. The HPD answers "which values are most plausible"; the equal-tailed answers "between which percentiles does the middle 95% sit". ## Reparameterisation This is the property that separates the two conceptually. Equal-tailed intervals are invariant under any increasing transformation: the 2.5th percentile of the rate maps to the 2.5th percentile of the log-odds, so transforming the endpoints of an equal-tailed interval gives the equal-tailed interval on the new scale. HPD intervals are not, because a change of variable rescales the density by a Jacobian factor, which reorders which values are "highest density". The HPD interval for a probability, transformed to log-odds, is generally not the HPD interval for the log-odds. If your team reports sometimes on the rate scale and sometimes on the odds or log-odds scale, that inconsistency is real. ## Computing them from posterior draws With posterior samples, the equal-tailed interval is just two sample quantiles - stable and trivially reproducible. The HPD is usually obtained by sorting the draws and sliding a window that spans 95% of them, keeping the narrowest window found. That estimator is noisier: it depends on the extreme ordering of the draws, so it needs more samples to settle, and on a multimodal posterior the sliding-window version silently returns one chunk when the honest answer is two. ## What to say in interview Define both precisely, note that they carry the same probability but different endpoints, give the skewed case as the example where the difference bites, and finish on the tradeoff: shortest versus reproducible and reparameterisation-stable. Saying "HPD is always better because it is shorter" misses the invariance and disjointness costs.