1 Shapley Value (Basic)

1.1 Basic Concept

Imagine a machine learning model that predicts whether a student will pass an exam (Yes/No) based on three factors:

The model’s baseline prediction (average chance of passing) is 50%.

Now, for a specific student, the model predicts 80% probability of passing.
SHAP values explain how each feature contributed to this prediction:

Feature	SHAP Value	Interpretation
Study Hours (+5 hours)	+20%	More study time increases passing probability.
Sleep Hours (+8 hours)	+10%	Enough sleep improves performance.
Distractions (Many distractions)	-10%	More distractions reduce the passing probability.

Baseline probability: 50%
SHAP contributions:
- Study Hours (+20%)
- Sleep Hours (+10%)
- Distractions (-10%)
Final prediction: 50% + 20% + 10% - 10% = 80%

Positive SHAP values push the prediction higher (increase passing probability).
Negative SHAP values push the prediction lower (decrease passing probability).
The sum of SHAP values explains why the model made its prediction.

This helps interpret the real impact of each feature on the model’s decision in a clear, human-readable way.

Output: Shapley value for the value of the j-th feature
Required: Number of iterations \(M\), instance of interest \(x\), feature index \(j\), data matrix \(X\), and machine learning model \(f\)
- For all \(m = 1, \dots, M\):
  - Draw random instance \(z\) from the data matrix \(X\)
  - Choose a random permutation \(o\) of the feature values
  - Order instance \(x\):
    \[ x_o = (x_{(1)}, \dots, x_{(j)}, \dots, x_{(p)}) \]
  - Order instance \(z\):
    \[ z_o = (z_{(1)}, \dots, z_{(j)}, \dots, z_{(p)}) \]
  - Construct two new instances:
    - With \(j\):
      \[ x_{+j} = (x_{(1)}, \dots, x_{(j-1)}, x_{(j)}, z_{(j+1)}, \dots, z_{(p)}) \]
    - Without \(j\):
      \[ x_{-j} = (x_{(1)}, \dots, x_{(j-1)}, z_{(j)}, z_{(j+1)}, \dots, z_{(p)}) \]
  - Compute marginal contribution:
    \[ \phi_j^m = \hat{f}(x_{+j}) - \hat{f}(x_{-j}) \]
Compute Shapley value as the average:
\[ \phi_j(x) = \frac{1}{M} \sum_{m=1}^{M} \phi_j^m \]

Let’s go through a step-by-step example of SHAP estimation based on the process in the image.

We have a simple dataset with 3 features and 4 data points.

Instance	Study Hours (X₁)	Sleep Hours (X₂)	Distractions (X₃)	Exam Score (f(x))
A (x)	5	7	2	?
B (z)	3	6	4	?
C	6	5	3	?
D	4	8	1	?

A simple model predicts exam scores as:
\[ f(X) = 10 \times X_1 + 5 \times X_2 - 3 \times X_3 \]

Choose the instance of interest
- Let’s compute SHAP for Study Hours (X₁) for Instance A (5 Study Hours, 7 Sleep Hours, 2 Distractions).
Draw a random instance \(z\) from data matrix \(X\)
- Randomly choose Instance B as \(z\) (3 Study Hours, 6 Sleep Hours, 4 Distractions).
Choose a random permutation \(o\) of feature order
- Let’s say the random order of features is (X₂ → X₁ → X₃).
Construct two new instances:
- Without \(X_1\) (Using \(z\)’s value for \(X_1\)):
  \[ x_{-1} = (7, 3, 2) \]
- With \(X_1\) (Keeping \(X_1\) from \(x\), others from \(z\)):
  \[ x_{+1} = (7, 5, 2) \]
Compute predictions for these instances:
- Using the formula:
  \[ f(X) = 10 \times X_1 + 5 \times X_2 - 3 \times X_3 \]
- Without \(X_1\):
  \[ f(x_{-1}) = (10 \times 3) + (5 \times 7) - (3 \times 2) = 30 + 35 - 6 = 59 \]
- With \(X_1\):
  \[ f(x_{+1}) = (10 \times 5) + (5 \times 7) - (3 \times 2) = 50 + 35 - 6 = 79 \]
Compute marginal contribution
\[ \phi_1^m = f(x_{+1}) - f(x_{-1}) = 79 - 59 = 20 \]
Repeat steps 2-6 for \(M\) iterations
- If we repeat this multiple times with different instances, we average the SHAP values.
Compute the final SHAP value as the average contribution:
\[ \phi_1 (x) = \frac{1}{M} \sum_{m=1}^{M} \phi_1^m \]
- If we performed 5 iterations, and the contributions were: 20, 22, 18, 21, 19,
- The final SHAP value is:
  \[ \phi_1(x) = \frac{20 + 22 + 18 + 21 + 19}{5} = 20 \]

This is how SHAP values fairly distribute the impact of each feature on the model’s prediction.