Representation of Strictly Proper Scoring Rule for Multiclasss Classificaiton

I am working on a classification problem, using features $$\mathbf{x}$$ to predict a target variable $$y \in \mathbb{N}_0$$. By a strictly proper scoring rule I mean a loss function $$\ell(y,\hat{y})$$ for which given any input $$\mathbf{x}$$, the unique minimizer of the expected loss is $$\operatorname{arg\,min}_{\hat{y}} \mathbb{E}_{y|\mathbf{x}} \left[\ell(y,\hat{y}) \right]=\mathbb{E}_{y|\mathbf{x}}[y|\mathbf{x}].$$ In other words, the loss incentivizes one to predict the expected value of the target variable, given the features. It is known that the squared error loss $$\ell(y, \hat{y})=(y-\hat{y})^2$$ satisfies this property. Furthermore, if we restrict the target variable to binary classification $$y \in \{0,1\}$$, then the Schervish representation theorem gives all strictly proper scoring rules as integrals involving a nonnegative weight function $$W$$: $$\ell(y,\hat{y}) = y \int_{\hat{y}}^1 (1-p) {W}(p) \mathrm{d}p + (1-y) \int_0^{\hat{y}} p {W}(p)\mathrm{d}p = \begin{cases} \int_{\hat{y}}^1 (1-p) {W}(p) \mathrm{d}p & y = 1 \\ \int_0^{\hat{y}} p {W}(p) \mathrm{d}p & y=0 \end{cases} .$$ For example $$W(p)=2$$ gives the squared error loss, while $$W(p)=(p(1-p))^{-1}$$ gives log loss.

I am interested in similar representations for more than 2 classes, particularly when the target variable domain is $$\mathbb{N}_0=\{0,1,2,\dots\}$$. Thanks.

• Would you kindly add a list of explicit questions? Are you looking for a Schervish representation theorem for multiple classes? Aug 30, 2023 at 19:54
• @Ggjj11 Yes, I want an analogue of the Schervish representation theorem for multiple classes. I.e. a characterization of all loss functions whose expected value is minimized at the conditional label mean. Ideally the set of labels I want is {0,1,2,...} as it contains all finite label sets as sub-cases. Aug 31, 2023 at 1:26
• As this is about mathematical statistics, the question might be a better fit on Cross Validated (Statistics) Stack Exchange, even if the topic isn’t strictly outside the purview of Data Science.
– Dave
Oct 4, 2023 at 23:00

$$((y(i)* \int_1^{y_i}(1-p) W(p) dp) + \sum_0 ^{ m-1}((1-y) \int_{y_i}^0 pW(p)/(m-1) dp))$$