Counterfactual Explanations

Intuition

"Your loan was rejected. If your income were 5 000 higher, it would have been approved." A counterfactual explains a decision by the smallest realistic change to the input that changes the outcome — and tells the person what they could do (recourse).

Precise explanation

For a classifier predict(X) -> labels, an instance $x$ with label $c$, and per-feature step sizes $\delta_j$ (0 = immutable, e.g. age or ethnicity):

• candidate changes: $x_j + k_j\delta_j$ with integers $|k_j| \le K$;
• cost: $\sum_j |k_j|$ (number of unit steps, an L1 distance in step units);
• find the minimum-cost candidate whose label differs from $c$, staying within optional bounds. Exhaustive search over $(2K+1)^{d_\text{mutable}}$ candidates is fine for small problems and gives the true minimum; real systems use optimisation or heuristics. Ties: smallest cost, then lexicographically smallest step vector $k$.

Worked example

predict = 1 if income + 2·savings ≥ 10; x = (4, 2) → 8 → class 0. Steps (1, 1): cost-1 options (5, 2) → 9 ✗, (4, 3) → 10 ✓ → counterfactual: savings + 1.

Common mistakes

• Changing immutable features.
• Returning any flipping point instead of the minimal one.
• Ignoring plausibility (bounds, realistic values).

Why this matters

Counterfactuals are human-friendly, actionable explanations and relate to legal discussions of a "right to explanation"; they also expose how close decisions are to the boundary.

References

• Interpretable Machine Learning (3rd edition) — Christoph Molnar (S024)