Proof: Brier scoring rule is strictly proper scoring rule
Theorem: The brier scoring rule is a strictly proper scoring rule.
Proof: We will show that both versions of the brier scoring rule (binary/multiclass) are strictly proper scoring rules.
1) Brier scoring rule for binary classification:
\[\label{eq:binary-bsr-s1} \mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] = - P(Y = 1) (q - 1)^2 + P(Y = 0) -q^2\]Let $p$ be the true probability of the event $Y = 1$. Then, the expected score is:
\[\label{eq:binary-bsr-s2} \mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] = - p (q - 1)^2 - (1 - p) q^2\]To find the maxima, take the derivative with respect to $q$ and set it to zero:
\[\begin{split} \frac{\partial}{\partial q}\mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] &= -2p(q - 1) - 2(1 - p)q \\ &= -2pq + 2p - 2q + 2pq \\ &= 2p - 2q \\ 0 &= 2p - 2q \\ \Rightarrow p &= q \end{split}\]We need to check the second derivative to see if it is a maximum (for the properness condition) and if it is the only maximizer (for the strictness condition):
\[\begin{split} \frac{\partial^2}{\partial q^2}\mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] &= -2 < 0 \\ \end{split}\]The second derivative is always negative which means that the function is concave and the maximum is unique. Therefore, $p = q$ is the only maximizer and the Brier scoring rule for binary classification is strictly proper.
2) Brier scoring rule for multiclass classification:
\[\begin{split} \mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] &= \sum_k P(Y = k) \bigg[ -\sum_i (q_i - y_i)^2 \bigg]\\ &= \sum_k P(Y = k) \bigg[ -(q_{k} - 1)^2 -\sum_{i \neq k} q_i^2 \bigg] \\ &= \sum_k P(Y = k) \bigg[ -(q_{k} - 1)^2 + q_k^2 -\sum_{i} q_i^2 \bigg] \\ &= \sum_k P(Y = k) \bigg[ -q_{k}^2 - 1 + 2q_k + q_k^2 -\sum_{i} q_i^2 \bigg] \\ &= \sum_k P(Y = k) \bigg[ 2q_k - 1 -\sum_{i} q_i^2 \bigg] \\ &= \sum_k P(Y = k)(2q_k - 1) - \sum_k P(Y = k) \bigg(\sum_i q_i^2\bigg) \\ &= \sum_k P(Y = k)(2q_k - 1) - \sum_i q_i^2 \bigg(\underbrace{\sum_k P(Y = k)}_\textrm{1}\bigg) \\ &= \sum_k P(Y = k)(2q_k - 1) - \sum_i q_i^2 \\ &= \sum_k P(Y = k)(2q_k - 1) - q_k^2 \end{split}\]Similar to what we did for log probability, this expression can be expressed as follows (replacing $q_K$ with $1 - \sum_{i \neq K} q_i$):
\[\begin{split} \mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] &= p_1(2q_1 - 1) - q_1^2 + p_2(2q_2 - 1) - q_2^2 + \ldots + p_K(2q_K - 1) - q_K^2 \\ &= p_1(2q_1 - 1) -q_1^2 + p_2(2q_2 - 1) -q_2^2 + \ldots + p_K \left( 1 - 2\sum_{i \neq K} q_i \right) - \left( 1 - \sum_{i \neq K} q_i \right)^2 \end{split}\]Taking the derivative with respect to $q_j$ and setting it to zero, we obtain:
\[\begin{split} \frac{\partial}{\partial q_j}\mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] &= 2p_j - 2q_j - 2p_K + 2 \left( 1 - \sum_{i \neq K} q_i \right) \\ &= 2p_j - 2q_j - 2p_K + 2 q_K \\ &= (p_j - q_j) + (q_K - p_K) \\ (p_j - q_j) &= (p_K - q_K) \\ \end{split}\]We know that $\sum_i p_i = 1$ and $\sum_i q_i = 1$, therefore:
\[\begin{split} p_1 - q_1 = p_2 - q_2 &= \ldots = p_K - q_K = \lambda \\ \sum_i p_i - q_i &= K \cdot \lambda = 0 \\ \Rightarrow \lambda &= 0 \quad \text{since} \; K \neq 0 \\ \Rightarrow p_i &= q_i \quad \text{for all} \; i = 1, \ldots, K \\ \end{split}\]Now, we need to check the second derivative to see, if it is a maximum for the properness condition and if it is the only maximizer for the strictness condition:
\[\frac{\partial^2}{\partial q_j^2}\mathbb{E}_{Y \sim P}[\mathbf{S}(Q, Y)] = - 2 - 2 = -4 < 0 \\\]The second derivative is always negative which means that the function is concave and the maximum is unique. Therefore, $p = q$ is the only maximizer and the Brier scoring rule for multiclass classification is strictly proper.
- Bálint Mucsányi, Michael Kirchhof, Elisa Nguyen, Alexander Rubinstein, Seong Joon Oh (2023): "Proper/Strictly Proper Scoring Rule"; in: Trustworthy Machine Learning; URL: https://trustworthyml.io/; DOI: 10.48550/arXiv.2310.08215.
Metadata: ID: P445 | shortcut: bsr-spsr | author: KarahanS | date: 2024-03-28, 20:40.