Interpreting Machine Learning Models with SHAP: A Comprehensive Guide

Date	Author	Version	Note
2024.06.20	Dog Tao	V1.0	Finish the document.

What is SHAP

SHAP (SHapley Additive exPlanations) values are a method used in machine learning to interpret the output of complex models. The SHAP value of a feature represents the impact that feature has on the prediction of a particular instance. It is based on concepts from cooperative game theory, specifically the Shapley value, which assigns a value to each player (or feature) in a way that fairly distributes the payout among them according to their contribution to the total payout.

Here are the key points about SHAP values:

1. Feature Contribution: SHAP values show how much each feature contributes to the prediction, either positively or negatively.
2. Additivity: The SHAP values for all features of a particular prediction add up to the difference between the model’s prediction and the average prediction over the dataset.
3. Consistency: If a model changes in a way that increases the marginal contribution of a feature, the SHAP value for that feature will not decrease.
4. Local Interpretability: SHAP values provide insight into the prediction for a single instance, helping to understand the model’s decision-making process for individual cases.
5. Global Interpretability: By aggregating SHAP values across many instances, you can gain an understanding of the overall importance of each feature in the model.

Mathematically, the SHAP value for a feature $i$ in an instance $x$ is calculated as follows:

ϕ i = ∑ S ⊆ F ∖ { i } ∣ S ∣ ! ( ∣ F ∣ − ∣ S ∣ − 1 ) ! ∣ F ∣ ! [ f ( S ∪ { i } ) − f ( S ) ] \phi_i = \sum_{S \subseteq F \setminus \{i\}} \frac{|S|!(|F| - |S| - 1)!}{|F|!} \left[ f(S \cup \{i\}) - f(S) \right] ϕi​=S⊆F∖{i}∑​∣F∣!∣S∣!(∣F∣−∣S∣−1)!​[f(S∪{i})−f(S)]
where:

• ${\varphi }_i$ is the SHAP value for feature $i$,
• $F$ is the set of all features,
• $S$ is a subset of features that does not include $i$,
• $f(S)$ is the prediction using the feature subset $S$,
• $|S|$ is the size of subset $S$,
• $|F|$ is the total number of features.

This formula considers all possible subsets of features and the change in the prediction when feature (i) is added to each subset, weighted by the size of the subsets.

SHAP values are widely used for their ability to provide consistent and interpretable explanations of model predictions, making them a valuable tool for understanding and debugging complex machine learning models.

Understanding Base Value

In SHAP (SHapley Additive exPlanations), the base value is a crucial concept that serves as a reference point for understanding the contribution of each feature to the prediction. Here’s a detailed explanation of what the base value means and its significance:

Definition of Base Value

The base value, often referred to as the expected value or mean prediction, is the average prediction of the model over the entire training dataset. It represents the starting point or the baseline from which SHAP values measure the contribution of each feature.

Significance of the Base Value

1. Reference Point for Interpretation: The base value acts as the reference point for the SHAP values. Each feature’s SHAP value shows how much that feature’s presence or value shifts the model’s prediction from this base value.
2. Model Explanation: By comparing the base value with the actual prediction for a specific instance, SHAP values explain the difference. The sum of all SHAP values for a particular instance, when added to the base value, equals the model’s prediction for that instance.

Contextual Examples

Regression Models

• Example: Suppose you have a model predicting house prices, and the base value is $300,000. This means that, on average, the model predicts a house price of $300,000 across all houses in the training dataset. For a specific house, if the model predicts $350,000, the SHAP values will explain how the features (e.g., number of bedrooms, location, etc.) contribute to increasing the prediction from $300,000 to $350,000.

Classification Models

• Example: For a binary classification model predicting whether a customer will buy a product (yes/no), the base value might be the average predicted probability of a customer buying the product, say 0.2 (or 20%). For a specific customer, if the model predicts a probability of 0.8 (or 80%), the SHAP values will show how each feature (e.g., age, income, browsing history) contributes to increasing the probability from 0.2 to 0.8.

Visual Representation

• Force Plot: In SHAP force plots, the base value is typically shown as a starting point on the left, with the contributions of individual features displayed as arrows pushing the prediction up or down from this base value. The sum of the base value and the SHAP values for all features gives the final prediction.
• Summary Plot: While summary plots primarily show the distribution of SHAP values for each feature, understanding the base value helps interpret how features generally impact predictions across the dataset.

Mathematical Context

In mathematical terms, if ( \phi_i ) represents the SHAP value for feature (i) and ( \phi_0 ) represents the base value, the prediction ( f(x) ) for an instance ( x ) can be expressed as:

$$f(x)={\varphi }_0+\sum _{i=1}^M{\varphi }_i$$
where:

• ${\varphi }_0$ is the base value (mean prediction).
• $M$ is the number of features.
• ${\sum }_{i=1}^M{\varphi }_i$ is the sum of the SHAP values for all features, representing their combined contribution to the prediction.

Understanding the base value in SHAP provides a foundation for interpreting how individual features influence the model’s predictions, making the model’s behavior more transparent and interpretable.

Understanding SHAP Value

Positive and negative SHAP values indicate how each feature influences the prediction of a machine learning model. Here’s an elaboration on the differences between positive and negative SHAP values in the contexts of classification and regression models:

Regression Models

In regression models, the goal is to predict a continuous outcome. SHAP values indicate how each feature influences the predicted value.

• Positive SHAP Values:

  • Interpretation: A positive SHAP value indicates that the feature increases the predicted value. This means the feature pushes the prediction higher than the baseline (average) prediction.
  • Example: For a model predicting house prices, if the feature “number of bedrooms” has a positive SHAP value, it means that having more bedrooms contributes to a higher predicted price.

• Negative SHAP Values:

  • Interpretation: A negative SHAP value indicates that the feature decreases the predicted value. This means the feature pushes the prediction lower than the baseline prediction.
  • Example: For the same house price prediction model, if the feature “distance from the city center” has a negative SHAP value, it means that being further from the city center contributes to a lower predicted price.

Classification Models

In classification models, the goal is to predict a categorical outcome. SHAP values indicate how each feature influences the likelihood of a particular class.

• Positive SHAP Values:

  • Interpretation: A positive SHAP value indicates that the feature increases the predicted probability of a specific class. This means the feature pushes the prediction towards that class.
  • Example: For a model predicting whether a loan will be approved or not, if the feature “income level” has a positive SHAP value for the “approved” class, it means that higher income increases the likelihood of the loan being approved.

• Negative SHAP Values:

  • Interpretation: A negative SHAP value indicates that the feature decreases the predicted probability of a specific class. This means the feature pushes the prediction away from that class.
  • Example: In the same loan approval model, if the feature “number of past defaults” has a negative SHAP value for the “approved” class, it means that having more past defaults decreases the likelihood of the loan being approved.

Visual Representation in Both Contexts

• Regression Models:

  • Force Plot: Shows how each feature’s SHAP value contributes to moving the prediction from the baseline to the final predicted value. Features with positive SHAP values push the prediction higher, while those with negative SHAP values push it lower.
  • Summary Plot: Displays the distribution of SHAP values for all features across all instances, illustrating which features generally increase or decrease the predictions.

• Classification Models:

  • Force Plot: Visualizes how each feature’s SHAP value contributes to the predicted probability of a specific class. Positive SHAP values push the probability towards the target class, while negative SHAP values push it away.
  • Summary Plot: Similar to regression, it shows the distribution of SHAP values for all features, indicating their overall impact on the predicted probabilities of the classes.

Understanding SHAP values in the context of classification and regression models helps in interpreting how features influence the model’s predictions, thereby enhancing the transparency and trustworthiness of machine learning models.