Your posterior for a parameter is bimodal — what goes wrong if you report only the posterior mean?

## What bimodality is telling you A posterior with two separated peaks is not noise; it is the analysis saying that two different parameter regions each account for the data well, and that the region between them accounts for it poorly. That is substantive information. Collapsing it to one number throws away exactly the part that was informative. ## Why the mean lands in the wrong place The posterior mean is a mass-weighted balance point. With mass piled at two separated locations, the balance point sits between them — in the valley. If the two modes are near-equally weighted, the mean sits close to the middle of the trough, which may be a region of very low posterior density. Nothing in the number itself signals this. A reader who receives only the mean will reasonably assume the posterior is a single hump centred there, and will act on a value the analysis considers among the less plausible. The median has a related problem: it is the value with half the mass on either side, which for two separated humps can again fall in the gap. Both summaries answer a question about location, and location is not what a two-humped posterior is about. ## The interval makes it worse, not better The standard reflex is to attach an interval. An equal-tailed interval — trim a fixed share of mass from each end — will typically stretch from inside the left mode to inside the right mode, covering the trough as though it were solid support. It reports the right total mass while implying the parameter is plausibly anywhere in between, which is precisely the opposite of what the posterior says. A highest-density region is honest here, because it can come out as a union of two disjoint intervals with a hole in the middle; the hole is the point. ## What to report instead Order of preference: 1. **The posterior itself.** A plot of the density is the fastest way to convey two competing explanations and costs one figure. 2. **The mass in each mode.** Something like: about two thirds of posterior mass sits near the lower region, one third near the upper. Two numbers, and the reader knows the whole story. 3. **The decision-relevant probability.** Most consumers of the analysis do not need the parameter, they need an action. The probability that the parameter exceeds a threshold, or that one option beats another, is a single number that stays meaningful no matter how strange the shape. 4. **A point estimate chosen from the loss, and labelled as such.** If a downstream system requires one number, derive it: the mean minimises expected squared error, the median minimises expected absolute error, and different losses can select different values. Reporting a mean and saying it is the squared-error-optimal action is defensible. Reporting a mean as if it were the plausible value of the parameter is not. ## Diagnose before you summarise Separated modes usually have a cause worth naming: - **Weak identifiability.** Two different parameter regions imply nearly the same observable predictions, so the data cannot separate them. More of the same data will not help; a different measurement or design might. - **Two regimes in the data.** A subgroup, a period, or a mixture whose components pull toward different parameter values. Modelling the structure explicitly usually replaces one confusing posterior with two clear ones. - **Symmetry in the model.** Some parameterisations admit an exchange that leaves the fit unchanged, producing mirror-image modes that are two labels for the same explanation. Here the fix is a constraint or a relabelling, not a summary. - **Prior-data conflict.** A prior concentrated somewhere the likelihood is not can leave mass in two separated places. That is a modelling conversation, not a reporting one. ## The senior habit The generalisable lesson goes beyond bimodality: always look at the posterior before choosing how to summarise it. Skew, heavy tails, hard boundaries at zero and modes sitting on a constraint all break the implicit assumption that a mean plus an interval is a faithful summary. The mean is a good default only when the posterior is roughly single-humped and symmetric — and you only know that by looking. Reporting summaries computed by habit and never inspecting the shape is a failure interviewers probe for deliberately, because it is the one that survives into production dashboards where nobody sees the density.