Why does minimising pinball loss at the 0.9 quantile give a higher forecast than minimising MAE?

## The loss function Pinball loss is defined for a target quantile level tau between 0 and 1. For an actual `y` and a forecast `q`: ``` if y >= q: loss = tau * (y - q) # the forecast fell short if y < q: loss = (1 - tau) * (q - y) # the forecast overshot ``` Both branches are non-negative and both are zero when `q = y`. The name comes from the shape: plotted against the forecast, the loss is a V with two different slopes, like the trajectory of a ball off a pinball flipper. Which slope is steeper is the entire content of the metric. At tau = 0.5 the two slopes are equal at 0.5, so the loss is exactly half the absolute error. Ranking forecasts by pinball loss at 0.5 and by mean absolute error therefore gives the same answer, and both are minimised by the median. Every other tau tilts the V. ## Why the minimiser is the quantile Think about moving the forecast up by one unit. Every future outcome that lies above the current forecast becomes one unit less short, saving tau per unit; every outcome below it becomes one unit more over, costing (1 - tau) per unit. Writing p for the probability that the actual exceeds the forecast, the expected change is `-tau * p + (1 - tau) * (1 - p)`. Raising the forecast is worth it while that is negative, and it stops being worth it exactly when `p = 1 - tau`. So at tau = 0.9 the optimum sits where the actual exceeds the forecast 10% of the time — the 90th percentile of the predictive distribution. At tau = 0.5 it sits where the actual is above and below equally often, which is the median. The asymmetric weights are not a fudge factor; they are precisely what pins the optimum to a chosen point of the distribution. ## Why this matters for a safety-stock forecast Suppose you hold stock for a part and want demand covered about nine times out of ten. The quantity you need is not the typical demand, it is the demand level that is exceeded only 10% of the time. A model tuned and scored on absolute error will hand you the median, and a median stock level runs out roughly half the time. The reverse mistake is just as common and harder to spot: a team produces a sensible 90th-percentile forecast and then evaluates it on the accuracy dashboard, which reports mean absolute error. The high forecast looks terrible there, because on most days it sits well above the actual and absolute error charges full price for that headroom. The forecast is then "improved" back toward the median and the stockouts return. Metric and target must agree; a quantile forecast has to be scored with the loss that defines it. The choice of tau itself is an economic decision, not a statistical one: it comes from the relative cost of being short versus being long. If a shortfall costs nine times what a unit of excess costs, the level that balances them is 0.9. Stating tau and stating why are part of the answer. ## Reading and reporting pinball loss Pinball loss is in the units of the series, and its magnitude also depends on tau, so a raw number carries no absolute meaning. Compare it only across forecasts of the same series at the same tau, or normalise it — for instance by the pinball loss of a naive benchmark producing the same quantile — before averaging across series. Averaging raw pinball losses over a catalogue lets the highest-volume series dominate. Averaging pinball loss across a grid of quantile levels for the same forecast gives a score of the whole predictive distribution rather than a single quantile, and as the grid gets finer that average approaches the continuous ranked probability score. That is the natural summary when a model emits a full distribution rather than one quantile. Pinball loss and interval coverage answer different questions. Coverage asks whether a stated level is honest; pinball loss is a proper score that rewards being honest and sharp at the same time, and unlike coverage it cannot be gamed by making the forecast extreme. A forecast pushed far above the 0.9 quantile never misses low, but it pays 0.1 per unit of excess on almost every period and its pinball loss gets worse, not better. ## What a strong answer sounds like Write the two branches, say which direction carries the larger weight at a high tau, and state the minimiser: the tau quantile, versus the median for absolute error. Then close the loop to the decision — the tau you pick encodes the cost asymmetry of the business problem, and scoring that forecast with a symmetric metric silently undoes it.