SHAP value

Raising a question

We are interested in understanding how each feature contributes to a prediction. In linear models, this is relatively straightforward to compute. Consider a linear model:

$$\hat{f}(x) = \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p$$

where $x$ is the input instance we want to explain, each $x_j$ for ( $j = 1, \ldots, p$ ) is the feature value, and ( $\beta_j$ ) is the coefficient corresponding to feature $j$.

The contribution of the $j$-th feature to the prediction ( $\hat{f}(x)$ ) is defined as:

$$\phi_j = \beta_j x_j - \mathbb{E}(\beta_j X_j) = \beta_j x_j - \beta_j \mathbb{E}(X_j)$$

Here, ( $\mathbb{E}(\beta_j X_j$) denotes the average contribution of the $j$-th feature. The contribution measures how much this feature deviates from its average impact. If we sum the contributions of all features for a given instance, we obtain:

$$\sum_{j=1}^p \phi_j = \sum_{j=1}^p (\beta_j x_j - \mathbb{E}(\beta_j X_j)) \\ = \left( \beta_0 + \sum_{j=1}^p \beta_j x_j \right) - \left( \beta_0 + \sum_{j=1}^p \mathbb{E}(\beta_j X_j) \right) \\ = \hat{f}(x) - \mathbb{E}(f(X))$$

In other words, the total contribution of input $x$ is equal to the prediction minus the average prediction.

Since we often work with non-linear models (e.g., ensemble methods), we require a more general approach. We adopt the Shapley value from cooperative game theory to attribute the prediction of any machine learning model to individual feature contributions.

The Shapley value of feature $j$ for an instance is defined as:

$$\phi_j(val) = \sum_{S \subseteq \{x_1, \ldots, x_p\} \setminus \{x_j\}} \frac{|S|!(p - |S| - 1)!}{p!} \left[ val(S \cup \{x_j\}) - val(S) \right]$$

Here:

• $S$ is a subset of the full set of features (a coalition),

• $x$ is the feature vector of the instance being explained,

• $p$is the total number of features,

• $\frac{|S|!(p - |S| - 1)!}{p!}$ is the Shapley weight for subset ( S ),

• $val(S)$ is the value function (i.e., model output) for feature subset $S$, marginalized over the features not in $S$. The value function $val_x(S)$ is computed as:

$$val_x(S) = \int f(x_1, \ldots, x_p) \, dP_{x_{\notin S}} - \mathbb{E}_X[\hat{f}(X)]$$

• This expression marginalizes over the features not included in $S$. For example, suppose the model uses 4 features $x_1, x_2, x_3, x_4$, and we consider the coalition $S = {x_1, x_3}$. Then the value function is:

$$val_x(S) = \int_{\mathbb{R}} \int_{\mathbb{R}} f(x_1, X_2, x_3, X_4) \, dP_{X_2, X_4} - \mathbb{E}_X[\hat{f}(X)]$$

SHAP specifies the explanation as:

$$g(\mathbf{z}') = \phi_0 + \sum_{j=1}^{M} \phi_j z'_j$$

where:

• $g$ is the explanation model,

• $\mathbf{z}' = (z'_1, \ldots, z'_M)^T \in \{0,1\}^M$is the coalition vector,

• $M$ is the maximum coalition size, and $\phi_j \in \mathbb{R}$ is the feature attribution for feature $j$, the Shapley values.

What is baseline?

在 TreeSHAP 中，baseline（有时也叫 expected value）是：

模型在训练集或测试集上所有预测值的平均值，即 $baseline=E[f(x)]$

• 对于回归问题：就是所有样本预测值的平均值。

• 对于二分类（log-odds 模型）：是 logit 输出的平均值。

• 对于概率输出（经过 sigmoid）模型：可以取平均概率作为 baseline（但 TreeSHAP 默认是在 log-odds 上计算）。

What is the SHAP additive?

SHAP 是加性的，但不代表模型是线性的

SHAP 的核心数学基础是 Shapley value，满足以下属性：

对于任意模型 $f(x)$，我们可以写成：

$f(x) = \phi_0 + \sum_{i = 1}^M \phi_i$

其中：

• $\phi_0$​ 是 baseline（即 $\mathbb{E}[f(x)]$）

• $\phi_i$​ 是第 $i$ 个特征对模型输出的边际贡献（SHAP 值）

SHAP value VS linear coefficient

• 在线性模型中，系数 $\beta$ 是模型本身学出来的参数。

  • 它反映的是每个特征单位变化对输出的直接影响。

  • 和特征标准化、缩放、共线性密切相关。

• 在 SHAP 中，$\phi$ 是模型预测后计算出来的归因值。

  • 它不是模型的参数，而是解释模型预测的“分摊结果”。

  • 会随着特征值、样本、模型结构、甚至特征顺序而变化。

假设你有一个模型预测：

💡 线性模型：

$\text{premium} = \beta_0 + 0.8 \cdot \text{age} + 2.5 \cdot \text{claim\_count}$

• 不论样本是年轻还是年老，age 的单位贡献永远是每多一岁 → 保费涨 0.8 元。

• 模型结构是“全局一致的”。

💡 非线性模型（如 XGBoost）+ SHAP：

• 对年轻人，age 的 SHAP 值可能是 -1.0 → 降低保费；

• 对年长的人，age 的 SHAP 值可能是 +3.0 → 提高保费；

• 说明模型对 age 的使用是非线性、有阈值的（比如 60 岁以上风险变大）。

Extra

• mean SHAP value 不等同于线性模型中的系数或贡献，它是对特征在整个样本空间中边际贡献的平均估计。虽然 SHAP decomposition 在形式上是加性的，但它能捕捉模型中复杂的非线性和特征交互，从而为任意模型提供一致的解释。

• The mean SHAP value reflects the feature’s average marginal contribution to the model output across all data points. A decrease in this value suggests that the model is now less reliant on this feature and more dependent on other variables to explain the predictions. However, this does not imply a consistent 15% contribution as would be the case in a linear model with fixed coefficients. For example, the SHAP value of a feature could account for 10% of the prediction in one data point, and 25% in another — highlighting the local, context-specific nature of SHAP-based explanations.

Reference

• https://christophm.github.io/interpretable-ml-book/shap.html
• https://zhuanlan.zhihu.com/p/85791430