Formula crosswalk
One forecast error can carry three multiclass numbers
Let K be the number of mutually exclusive outcomes. The original multicategory form averages the sum of K squared class errors over N rows. Other conventions divide that same value by K or by two. The conversion is arithmetic, not a performance claim.
Exactly one y value is 1 for each settled row; all other class indicators are 0.
| Declared convention | Relationship to SΣ | Theoretical range | Same 1X2 error |
|---|---|---|---|
| Summed full vector | SΣ | 0 to 2 | 0.450 |
| Class-averaged | SΣ ÷ K | 0 to 2/K | 0.150 |
| Half-scaled | SΣ ÷ 2 | 0 to 1 | 0.225 |
| Binary event MSE | SΣ ÷ 2 only when K = 2 | 0 to 1 | Not a 1X2 convention |
For uniform K-class probabilities, the summed reference is 1 − 1/K. In 1X2 that is 0.6667 summed, 0.2222 class-averaged or 0.3333 half-scaled.
Search question
Is a Brier score of 0.15 better than 0.45?
If 0.15 is a class-averaged three-outcome score and 0.45 is the original summed score on the same rows, they are identical. If both use the same convention, 0.15 is lower—but “lower” becomes an admissible model comparison only after matching the exact targets, rows, exclusions, weights and aggregation.
Interpretation boundary
What is a good Brier score for football predictions?
There is no context-free threshold. Lower is better only within a fixed scoring convention and outcome task. A useful report compares the model with a frozen baseline on identical rows, publishes the sample size and period, and inspects calibration instead of promoting one aggregate number.
- 01
Fix the task
1X2, BTTS and binary match events have different class structures and base rates.
- 02
Fix the scale
Name the formula and K. “Brier 0.20” alone is not reproducible.
- 03
Fix the rows
Use the same settled matches, exclusions, time cut-off and row weights.
- 04
Fix the baseline
Compare against a declared frozen reference rather than an invented universal pass mark.
Do not auto-convert
Four labels that need more than a multiplier
Brier Skill Score
BSS = 1 − BS/BSref. Without the exact reference score and matched rows, it cannot be reversed into a raw Brier score.
Percentage-point squares
Probabilities must be divided by 100 before applying the standard formula. Squaring percentage points produces values 10,000 times the decimal-space score.
RPS, log loss or ECE
These answer different questions. None can be recovered from an aggregate Brier score by a fixed conversion.
Unknown convention
A plausible-looking number is not enough to infer a formula. Ask the publisher for the equation, K, sample and exclusions.
Precision-aware comparison
0.150 is a rounded interval, not an infinitely precise value
The checker preserves the typed decimal places. A displayed value of 0.150 represents the rounding interval from 0.1495 to 0.1505 before scale conversion, assuming ordinary rounding to the nearest displayed unit. Two converted point estimates may differ while their declared precision intervals still overlap. The JSON passport records that distinction.
Interval overlap means arithmetic compatibility at the displayed precision. If a publisher truncated instead of rounded, its rule must be checked separately. Compatibility does not establish calibration, statistical significance, independence, profitability or historical publication.
Primary and authoritative sources
Definitions anchored to the scoring-rule literature
- Brier (1950), verification of probabilistic forecasts.Original paper and DOI
- Gneiting and Raftery (2007), strictly proper scoring rules.Journal article and DOI
- scikit-learn, current Brier score loss convention and scale_by_half behavior.Authoritative API documentation