Permutation Feature Importance

Intuition

If shuffling a feature's column (destroying its relationship with the target) makes the model much worse, the model relied on that feature. If nothing changes, the feature didn't matter to this model.

Precise explanation

1. Baseline score $s_0 = \text{score}(y, f(X))$ (higher = better, e.g. R² or accuracy).
2. For each feature $j$ and repeat $r$: permute column j (rows shuffled with rng.permutation(n)), compute $s_{j,r}$.
3. Importance $I_j = s_0 - \text{mean}_r\, s_{j,r}$, with standard deviation across repeats. Use one generator default_rng(seed), iterating features in order and repeats inside. Small negative values mean "no importance" (noise). Correlated features share importance — permuting one can be compensated by the other. Computing on held-out data measures importance for generalisation.

Worked example

y = 3x₀ + noise, x₁ pure noise, linear model: permuting x₀ drops R² from ≈1 to ≈−1; permuting x₁ ≈ 0.

Common mistakes

• Re-fitting the model after permuting (that is drop-column importance, a different method).
• Permuting all features at once.
• Reporting a single permutation without variability.

Why this matters

Model-agnostic, cheap and widely used (scikit-learn permutation_importance); it also connects to the permutation test idea from ST-20.

Library implementation

sklearn.inspection.permutation_importance(estimator, X, y, n_repeats=..., random_state=...) implements the same algorithm for fitted scikit-learn models (and pipelines). It uses its own random shuffles, so individual numbers differ slightly from yours, but the ranking of clearly important features agrees. The comparison runs when scikit-learn is installed.

References

• Interpretable Machine Learning (3rd edition) — Christoph Molnar (S024)
• User Guide (scikit-learn 1.9.1) — scikit-learn developers (S012)