When is the posterior mode a poor summary of a skewed posterior distribution?

## What a point summary is being asked to do A posterior is a whole distribution. Compressing it to one number is a decision, and different compressions answer different questions. The three standard summaries are: - the **mode**, the value where the posterior density is highest — this is the MAP estimate; - the **median**, the value with half the posterior mass on each side; - the **mean**, the probability-weighted average of the parameter. For a symmetric unimodal posterior all three coincide, and the choice is free. Skew is what separates them, and it is exactly where the choice starts to matter. ## A concrete skewed case Suppose the posterior for a latency parameter has a Gamma shape with shape parameter 3 and rate 1. Then: - mode = `(shape - 1) / rate` = 2 - median is approximately 2.67 - mean = `shape / rate` = 3 The ordering `mode < median < mean` is the signature of right skew, and it is worth being able to state on sight: the long right tail drags the mean up, the mode stays back at the peak, and the median sits between them. Three summaries, three answers, one posterior. Anyone who reports a single number without saying which summary it is has left the reader unable to reconstruct what was claimed. ## The decision-theoretic account The principled way to pick is to name the loss function. If `theta` is the truth and `a` is the number you report, Bayesian decision theory says to choose the `a` that minimises the posterior expected loss. - Squared-error loss, `(theta - a)^2`, is minimised by the **posterior mean**. - Absolute-error loss, `|theta - a|`, is minimised by the **posterior median**. - An all-or-nothing loss that charges a fixed penalty unless `a` is essentially exactly `theta` is minimised, in the limiting sense, by the **posterior mode**. That last loss is the one almost nobody has. Reporting a latency parameter, a conversion rate or a coefficient is not a game where being close counts for nothing. So the mode's popularity comes from computational convenience — it is the answer optimisation gives you, without any integration — rather than from a decision anyone was actually making. ## Asymmetric costs Many real problems have costs that are not symmetric at all. If under-provisioning a service costs far more than over-provisioning it, then no measure of centre is the right report. The loss-minimising action sits deliberately off-centre — for a strongly asymmetric loss, at an upper quantile of the posterior. Answering *which point summary should I use* with *neither, because your loss is asymmetric and the answer is a quantile* is the response that separates a senior candidate from a competent one. ## Why the mode gets worse as dimensions grow In a one-dimensional problem the mode at least sits in a region of appreciable probability. In many dimensions this stops being true. Density and mass come apart: the highest-density point occupies a vanishingly small volume, while the bulk of the posterior mass lives in a shell some distance away from it. A parameter vector drawn from the posterior will typically look nothing like the mode — its norm will be larger, its components less extreme in aggregate. So the MAP vector can be an untypical draw from the very distribution it summarises, and predictions made by plugging it in can be systematically different from predictions averaged over the posterior. ## Other failure modes to name - **Sharpness is invisible.** Two posteriors with the same mode can have wildly different spreads. The mode alone cannot distinguish a decisive result from a shrug. - **Boundaries.** When the posterior peaks at the edge of the parameter space — a variance at zero, a probability at one — the mode sits on the boundary and reports a degenerate answer that the mass of the posterior contradicts. - **Density units.** The mode is the argmax of a density, and a density is not a probability; it depends on how the parameter was scaled. The mean and the median depend on the distribution's mass rather than its peak height, which is why they behave more stably. ## What to do instead Report the summary that matches the decision, and say which one it is. When the posterior is skewed or the audience will act on the number, hand over more than a point: the shape, or at least a second summary that reveals the skew. If a single number is genuinely required, derive it from the loss the number will be used against rather than from whichever quantity was easiest to compute.