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

Uncertainty term — where it appears

The variance of the events themselves, which every forecaster in a given world is charged and none can reduce. It is the score a forecaster achieves by issuing the base rate every time, so it is the benchmark that a skill score divides by and the ceiling any comparison is read against.

Named by 3 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 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

Named alongside it

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

Base rateBrier scoreForecast calibrationProbability forecastQuadratureBinningClimatological forecastLogarithmic scoreMurphy's decompositionProper scoring ruleReliability termWithin-bin variance

All concepts