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

Proper scoring rule — where it appears

A rule for charging a probability forecast whose expected charge is smallest when the forecaster reports what it actually believes. It is what makes a score a measurement rather than an incentive, and it is a property to be checked by finding the optimum rather than assumed from the shape of the formula.

Named by 4 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 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 One curve, and two forecasters that are each a single point. The ROC curve of an honest forecaster whose signal has correlation 1 with the latent state — every threshold on its probability, from the bivariate normal — and two forecasters that only ever say 0 or 1. The one an absolute-error score pays for says 1 wherever the honest probability exceeds a half: it has a true-positive rate of 0.6742 and a false-positive rate of 0.1375, and its "curve" is the two straight segments through that point, with area (TPR + TNR)/2 = 0.7684. The honest curve's area is 0.8683. Thresholding at the base rate of 0.3744 instead of at a half gives the largest area any two-valued forecaster can have here, 0.7818, and it is still 0.0866 short.

The liar with two answers

The forecaster an absolute-error score pays for says only 0 or 1, and on the ROC square it is a single point: its area is (TPR + TNR)/2 = 0.7684, against the honest forecaster's 0.8683, and it falls below the honest one on 200 of 200 counted records. No relabelling of its two answers returns what it threw away — the best recovers a Brier score short of the honest one by exactly the 0.022154 of resolution lost — and below a signal correlation of 0.7332 the same score prefers saying no every time to an honest forecast.

calibrate · Calibration The honest curve, and the same forecaster in three coarse vocabularies. The ROC curve of the honest probability at full signal, area 0.8683, beside the same forecaster rounded to the nearest whole number, area 0.7684 — the two-valued liar — to the nearest half, area 0.8205, and to the nearest tenth, area 0.8650. A vocabulary of v values is v points on the square joined by straight segments, and tied reports count half.

A forecaster that rounds

An honest probability issued in tenths loses 0.0033 of ROC area and 0.000708 of resolution — the variance its bands average away, and 89.5% of the 0.000792 it adds to the Brier score. Two hundred records of two thousand forecasts show that loss on 189; it takes about 3,300 forecasts to put it two standard errors from zero. And 3.207 in every thousand forecasts in tenths are a 0% on an event that happened, which a logarithmic score charges without limit.

calibrate · Calibration

Named alongside it

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

Brier scoreProbability forecastBase rateDiscriminationMurphy's decompositionReliability termResolution termROC curveClimatological forecastClosed formForecast calibrationImproper scoring rule

All concepts