Concept

Resolution term — where it appears

The part of a score that credits a forecaster for telling one case from another: how far the event rates in its own groups spread away from the overall base rate. It is zero for a forecaster that issues the same number every time, and it is what separates a useful forecaster from a merely calibrated one.

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

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

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

Where three vocabularies of 5 values put their cuts, under the honest forecasts they divide. The density of the honest probability at full signal, piled toward small values because the event happens 37.4% of the time, and below it three ways of cutting it into 5 groups, each printing its group's event rate. 5 equal-width bins: cuts at 0.200, 0.400, 0.600, 0.800, printed values 0.075, 0.295, 0.497, 0.698, 0.904, resolution lost 0.003454. 5 equal-mass groups: cuts at 0.066, 0.211, 0.422, 0.695, printed values 0.025, 0.132, 0.311, 0.554, 0.850, resolution lost 0.004046. the best 5: cuts at 0.145, 0.339, 0.552, 0.777, printed values 0.054, 0.236, 0.441, 0.662, 0.892, resolution lost 0.003193. The density is cut off at 3.2 near zero, where it grows without bound. The best vocabulary's cuts sit halfway between neighbouring printed values, and its printed values sit at their groups' event rates; no other partition into 5 groups loses less.

Five values chosen by the forecasts

The vocabulary of five calibrated values that loses least on the Brier score puts its cuts at 0.145, 0.339, 0.552 and 0.777, and gives up 0.003193 of resolution against 0.003454 for equal-width bins and 0.004046 for equal-mass groups. Eleven such values beat tenths by 18.5% — but only once someone knows where to cut. Fitted on a record of two thousand forecasts the same eleven add 0.001547, about twice what tenths add with nothing fitted, and they draw level only near twelve thousand. For users acting at chances near one in ten, a vocabulary cut for them lowers their regret by a factor of 7.5 and costs the Brier score nearly nine times as much.

calibrate · Calibration

Named alongside it

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

Brier score Probability forecast Reliability term Discrimination Murphy's decomposition Proper scoring rule Base rate Closed form ROC curve Climatological forecast Forecast calibration Monte Carlo

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