# intuition behind Brier score

Assume that we have some count data $$x_{1}, \dots, x_{n}$$, which take values $$\{1, \dots, m\}$$ and we have some estimator of the probability mass function, $$\hat{\mathbf{p}} = (\hat{p}_{1}, \dots, \hat{p}_{m})$$. In order to evaluate the performance of the estimator one can use Brier score which is the following. Let $$\mathbf{I}_{i} = (0, \dots, 1, \dots, 0)$$ is a vector in $$\mathbb{R}^{m}$$, with value $$1$$ is at the $$t$$-th position from the beginning of the vector, if $$x_{i} = t$$, for $$t \in \{1, \dots, m\}$$ and all $$i=1,\dots, n$$ . The Brier score is given by $$BS = \frac{1}{n}\sum_{i=1}^{n}||\mathbf{I}_{i} - \hat{\mathbf{p}}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{n} \sum_{j=1}^{m}(I_{i,j} - \hat{p}_{j})^2.$$

I have got the following questions: what is the intuition behind this score? Will BS converge (in some sense) to the smallest value if (iff?) the estimator is consistent?

Is this basically a quadratic score, suggested by Stone in

M. Stone (1974) Cross-Validation and Multinomial Prediction Biometrika, Vol. 61, pp. 509-515 ?

How common is the use of it in practice?