Explaining a Linear Prediction

Intuition

A linear model is a weighted sum, so its prediction splits naturally into one piece per feature. But "feature 2 contributed +30" only makes sense relative to something: compared with an average customer, a zero input, or last month's value.

Precise explanation

For $f(x) = w\cdot x + b$ and a baseline (reference) input $x'$ (often the training mean):

The second property — contributions add up exactly to the difference from the baseline prediction — is called efficiency (or local accuracy). Using $\phi_j = w_jx_j$ (baseline 0) is a special case and can be misleading when 0 is not a meaningful input (age 0, temperature 0 K). Interpretation also depends on feature scale: standardised features make $w_j$ comparable.

Worked example

w = [2, −1], b = 5, x = [3, 4], baseline = mean [1, 1]: φ = [4, −3]; f(x) = 7, f(x′) = 6, sum φ = 1 = 7 − 6 ✓.

Common mistakes

• Reporting $w_j$ alone as "importance" for one prediction (ignores the input value).
• Forgetting the baseline: contributions then do not add up to anything interpretable.
• Comparing weights of unscaled features.

Why this matters

This is the simplest exact attribution; Shapley values (XA-11) reduce to exactly this formula for linear models with independent features.

References

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