> For the complete documentation index, see [llms.txt](https://ai.younglimit.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ai.younglimit.com/responsible-ai/xai/shap-value.md).
# 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
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