A posterior is a whole distribution — every value gets a plausibility. But often you’re forced to report just one number: “how many days until this bug is fixed?”, not “here’s a distribution over days.” Three standard candidates:
- Mean: the probability-weighted average. Sensitive to a long tail — a few very-late outcomes pull it upward even if most outcomes are quick.
- Median: the 50th percentile — the value such that half the posterior’s probability lies below it, half above. Ignores how far the tail extends, only how much probability is out there.
- Mode: the single most plausible value — the peak of the distribution. Ignores everything about the shape except where the peak sits.
These aren’t three ways of approximating the “same” right answer — each one is the number that minimizes a different penalty (loss function) for being wrong, and they can genuinely disagree, especially on a skewed distribution.
Your posterior for “days until this bug is fixed” is right-skewed (most fixes are quick, but a long tail of nasty ones drags upward): mean = 8 days, median = 6 days, mode = 4 days.
The rule connecting loss to estimate (you don’t need to prove it, just use it):
- If you’re penalized by squared error —
(guess − actual)²— the mean minimizes your expected penalty. - If you’re penalized by absolute error —
|guess − actual|— the median minimizes it. - If you only get credit for guessing the exact right value (0/1 loss — right or nothing) — the mode minimizes it.
Given your bonus structure above (linear penalty, proportional to absolute distance, symmetric in both directions), which point estimate should you report?