Strictly proper scoring rules on a finite label set

For an integer $K\ge 2$, let the label set be $\mathcal{Y}=\{1,\dots ,K\}$ and let ${\mathrm{\Delta }}_K=\{q\in [0,1{]}^K:{\sum }_{j=1}^K{q}_j=1\}$. A finite-label score loss is a function $S:{\mathrm{\Delta }}_K\times \mathcal{Y}\to [0,+\mathrm{\infty }]$ whose section $q\mapsto S(q,j)$ is Borel measurable for every label $j$. If the true label distribution is $p\in {\mathrm{\Delta }}_K$ and the reported forecast is $q\in {\mathrm{\Delta }}_K$, its expected score loss is $\bar{S}(p,q)={\sum }_{j=1}^K{p}_{j}S(q,j)$, where a term with $p_j=0$ contributes zero even when $S(q,j)=+\mathrm{\infty }$. The loss is proper if $\bar{S}(p,p)\le \bar{S}(p,q)$ for every $p,q\in {\mathrm{\Delta }}_K$, and strictly proper if it is proper and equality implies $q=p$. The logarithmic loss is $S_{\log }(q,j)=-\log q_j$, with $-\log 0=+\mathrm{\infty }$, where $\log$ is the natural logarithm; the Brier loss is $S_{\mathrm{B}}(q,j)={\sum }_{i=1}^K(q_i-{\mathbf{1}}_{\{i=j\}}{)}^2$.

Loss functions and risk: a loss is nonnegative and measurable in its action for each fixed parameter; no measurability on the parameter space is required, and its risk is the corresponding extended nonnegative integral.

The natural logarithm as the inverse of the exponential function: $\log$ is the inverse of the exponential function on positive reals.

The Borel sigma-algebra of a topological space: Borel subsets are the sets in the sigma-algebra generated by the open subsets of a topological space.

Wikipedia, Scoring rule: Definitions and Proper scoring rules specify forecasts, expected score, loss orientation, properness, and strictness; Categorical variables gives the finite-label logarithmic and Brier examples. The measurable extended nonnegative-loss conventions are stated explicitly here; the logarithmic reward example is negated.