= Brier score
{c}
{title2=$L_{\rm B}(p,y)=(p-y)^2$}
{wiki}
= Brier loss
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{synonym}
The binary <Brier score> applies <squared-error loss> to a reported success <probability> $p$ and the observed indicator $y\in\{0,1\}$. It is a loss, so smaller is better. Its values lie in $[0,1]$: predicting the outcome with certainty gives zero, while assigning it zero <probability> gives one. Unlike the <logarithmic scoring rule> loss, the binary <Brier score> remains bounded even for a confidently wrong forecast.
For a <Bernoulli distribution> with true success <probability> $q$, averaging the two squared losses gives
$$
\mathbb E_q[(p-Y)^2]=q(p-1)^2+(1-q)p^2=(p-q)^2+q(1-q).
$$
The report controls only $(p-q)^2$; the remaining term is the <variance> of the <Bernoulli random variable>. Consequently the <Brier score> is a <strictly proper scoring rule>. At $q=0.7$, the expected loss is $(p-0.7)^2+0.21$, with unique minimum $0.21$ at $p=0.7$. Reporting $p=0.2$ instead gives expected loss $0.46$, despite its better loss on the particular outcome $Y=0$ above.
For several possible categories, the <Brier score> sums the squared differences between reported <probabilities> and observed category indicators. With two categories that sum is $2(p-y)^2$, so its numerical values are twice the binary convention used here, with the same optimizing report. Brier's original https://www.u-cursos.cl/ingenieria/2025/1/CI71C/1/material_docente/bajar?bajar=1&id=8745481[Verification of Forecasts Expressed in Terms of Probability (1950), equation (2)]{ref} averages these category-sum losses over forecasts. <Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others> (Section 3.1, Example 1, and Table 1) gives the category-sum and binary forms as negative rewards; negating them gives the losses described here.
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