Normal Distribution
nothing is called 95% until it has been counted
Concept

Forecast calibration — where it appears

Whether a probability forecast means what it says: of the occasions a forecaster said 30%, about three in ten should have happened. It is a property of a record rather than of any one forecast, it is checked by grouping forecasts and counting, and it is satisfied exactly by a forecaster that issues the base rate every time.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

What each forecaster says, and what is true. The true probability of the event given a forecaster's signal, and what three forecasters report. The truth is Φ(-0.5 + 1.2w) and the honest forecaster reports it, so its curve and the truth are the same line. The loud forecaster pushes every probability towards the ends and the hedged one pulls every probability towards the middle; both have their mean report held at the base rate of 0.3744, so each crosses the truth exactly once and neither can be caught by checking its average. Their reliability terms are 0.008500 and 0.013025 against the honest forecaster's zero, and all three have the same area under the ROC curve, 0.868312.

An identity in three terms

Reliability minus resolution plus uncertainty is quoted as a rewriting of a probability score. It is an identity to 2.6·10⁻¹⁵ on the one grouping where reliability is the whole score and resolution exactly cancels uncertainty, and it is out by 0.004125 on the coarsest grouping anybody would actually draw.

calibrate · Calibration One forecaster, one sample, and the bins it was read in. A reliability diagram of 500 forecasts in 10 equal-count bins, drawn over the curve the same forecaster has in population. The forecaster is honest, so its population curve is the diagonal exactly and every departure the points show is sampling. The sample's reliability term reads 0.002262 and its expected calibration error 0.0358, against a true reliability of 0.000000. Both are properties of this binning as much as of this forecaster: at 5 bins and at 50 the same honest forecaster reports 0.001429 and 0.014100.

A curve that is a binning

A forecaster with no miscalibration in it at all reads 0.001429 at five bins and 0.014100 at fifty, on the same five hundred forecasts. The closed form is K/n times the forecaster's own irreducible score, and subtracting it returns zero.

calibrate · Calibration Six forecasters, all calibrated, not equally useful. The resolution of six forecasters that are all perfectly calibrated, each reporting the true probability of the event given a signal that carries more or less of the latent state. The largest reliability anywhere in the family is 2.0e-33, so a calibration check passes every one of them. They are not equally good: resolution runs from exactly 0.00 for the forecaster that issues the base rate every time to 0.092758 for the one that sees everything, and their Brier scores run from 0.234237 — which is the world's own uncertainty, and the score of a table of base rates — to 0.141479. Calibration is a necessary condition that a constant forecast satisfies exactly.

Calibrated and useless

Six forecasters that are calibrated to 2·10⁻³³ run from resolution exactly 0 to 0.092758, and three forecasters with reliabilities from 0 to 0.013025 have areas under the ROC curve identical to every bit a double carries. Each measure is exactly blind to what the other one sees.

calibrate · Calibration Where each rule says to put the number. The expected score of reporting each probability on the axis when the event's true probability is 0.25, for four scoring rules. The Brier, logarithmic and spherical scores each bottom out at 0.25 — found by search rather than assumed, to 8 decimal places — which is what makes them proper: a forecaster with a genuine belief cannot improve its expected score by reporting anything else. The absolute-error score is a straight line in the reported value, p + r(1 − 2p), so it has no interior minimum at all; its optimum is 0.0, a distance of 0.250 from the truth, and taking it saves 0.125.

A score that rewards lying

An absolute-error score pays a forecaster exactly ⅛ of a point to replace a true quarter with a zero, and over two hundred records a liar beats a truthful forecaster on 200 of 200. A skill score against the forecaster's own average buys 0.012633 of reported skill for 0.002035 of real score.

calibrate · Calibration How long an honest forecaster looks broken for. The calibration error shown by a forecaster with no miscalibration in it at all, at five record lengths, drawn against one over the square root of the length so that the closed form is a straight line through the origin. It is 0.1252 at fifty forecasts and 0.0090 at ten thousand, against a closed form of √(2K/πn) times the mean root bin variance which gives 0.1257 and 0.0089. The threshold drawn across it is 0.02, a figure routinely read as evidence that something is wrong; the mean falls under it at 1976 forecasts and the 95th percentile at about 4111. Below that, an honest forecaster and a miscalibrated one are being told apart by a statistic that is mostly the sample size.

The miscalibration a perfect forecaster shows

A forecaster whose true reliability is exactly zero shows a calibration error of 0.1252 on fifty forecasts and 0.0090 on ten thousand. Every one of 1,200 blameless hundred-forecast records exceeds the 0.02 routinely read as evidence of a problem, and the mean does not fall under it until 1,976 forecasts.

calibrate · Calibration

Named alongside it

The objects these essays reach for when they reach for this one.

Probability forecastBase rateBrier scoreClimatological forecastReliability termBinningQuadratureUncertainty termDebiased estimatorDiscriminationExpected calibration errorLogarithmic score

All concepts