xai 2

1. marginal contribution
   • contribution of an element j to a function f when j is added, holding everything else fixed

• 4 shapley axioms:
  • efficiency - the shapley values sum to the gap between full coalition and empty one
  • symmetry - if v(S U {j}) = v(S U {k}) for S that excludes both then MC_j= MC_k
  • dummy - if v(S U {j}) = v(S) for every S then MC_j= 0
  • additivity - for split in value function, the players shapley values also split MC_J(v+w) = MC_j(v) + MC_j(w)
• Uniqueness theorem - shapley values satisfy all 4 axioms so it is probably fair!
• computational challenge - since there are 2^p coallitions for mc_j exact computation is infeasible!
• shapley values
  • strengths:
    • axiomatic - satisfies axioms
    • model-agnostic - works for any value function
    • interpretable - MC_j has a per player meaning
  • limitations:
    • cost - 2^p coalitions
    • missing-feature semantic - feature absent not possible in ML
    • correlated features - credit can leak to features not used by models

2. SHAP

• we need a data matrix “X” - because in ML we can’t skip features, to evaluate coalition S, we plug in stand-in values for feat not is S, drawn from X
• SHAP - Shapley Additive exPlanations
• explains individual predictions of an ML model using Shapley values
  • Shapley - uses shapley values
  • Additive - output = baseline + each feat contribution
  • exPlanation - local explanation properties of SHAP:
• post-hoc → analyzes the model after it has already been trained.
• model-agnostic → works with any architecture - xgboost, nn, random forest, svm etc.

SHAP axioms:

• local accuracy - baseline + sum of feat contribution = model pred
• missingness - if feat j is missing from input then mc_j=0
• consistency - changing model such that feat j’s marginal contribution increases or decreases but SHAP guaranteed it will never decrease

example:

• step-1: is to calculate the baseline!
• step-2: find the marginal contribution for all the feat except the target feat so here we want to find out the MC for BP but we find out the value for Age first
• now we include BP as well and we can get the MC for BP = v(Age, BP) - v(Age)
• similarly find out the mc for all feat combinations except BP and then take the weighted avg!

Links:

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