SHAP Values¶

Formula¶

\[ \phi_i = \sum_{S\subseteq F\setminus\{i\}} \frac{|S|!(|F|-|S|-1)!}{|F|!}\big(v(S\cup\{i\})-v(S)\big) \]

Parameters¶

\(\phi_i\): contribution of feature \(i\)
\(F\): set of features
\(v(S)\): model value for coalition \(S\)

What it means¶

SHAP values attribute a prediction to features using a game-theoretic additive decomposition.

What it's used for¶

Local explanation of individual predictions.
Global summaries by aggregating absolute SHAP values.

Key properties¶

Additive feature attributions sum to prediction difference from a baseline.
Practical implementations use approximations for speed.

Common gotchas¶

Values depend on background/reference distribution choice.
Correlated features can split credit in unintuitive ways.

Example¶

For one loan application, SHAP can show high debt ratio pushing risk up while long employment pushes it down.

How to Compute (Pseudocode)¶

Input: trained model f, example x, background/reference data, SHAP method
Output: SHAP values phi for x

choose a SHAP algorithm appropriate for the model family
  examples: TreeSHAP for tree ensembles, KernelSHAP for model-agnostic approximation
compute baseline value from the background/reference data
estimate or compute feature contributions phi so that:
  baseline + sum(phi_i) approximates (or equals) the model output for x
return phi

Complexity¶

Time: Depends heavily on the SHAP method and model family (exact Shapley computation is exponential in the number of features; practical SHAP implementations use specialized algorithms or approximations)
Space: Depends on the method, model, and background sample size
Assumptions: This card describes the general SHAP workflow; TreeSHAP, KernelSHAP, and sampling-based methods have very different runtime/memory profiles