SHAP for real models

A linear model predicting a house's price from four features. Every bar is that feature's SHAP value for this specific house — and unlike the pizza chapter, none of these came from averaging over $4! = 24$ orderings. For a linear model, there's an exact shortcut.

For a linear model $f(x) = b + \sum_i w_i x_i$, the Shapley value has a closed form — no permutations needed at all:

• $\phi_i$ — feature $i$'s SHAP value: its contribution to this specific prediction.
• $w_i$ — the linear model's weight (coefficient) for feature $i$.
• $x_i$ — this instance's actual value for feature $i$.
• $\bar x_i$ — feature $i$'s baseline value, typically its average across some reference population.

1. Credit is weight times distance from typical

   A feature's credit is just its weight times how far this instance sits from "typical" — exactly what you'd guess a fair split should look like.

2. Provably matches the full permutation average

   For a linear model specifically, this closed form is provably the same answer the full permutation average from the pizza chapter would have given.

Drag the size slider. Only size's own bar moves — every other feature's SHAP value is untouched, because in a linear model each feature's contribution genuinely doesn't depend on any other feature's value. Watch the total, though: baseline plus every bar summed still always equals the exact prediction, at any size you drag to.

Baseline (a typical house): size $20$, age $15$, distance $10$, not renovated — predicting $320$. This house: size $25$, age $5$, distance $8$, renovated — predicting $445$:

1. Size's contribution

   $w_{\text{size}}(25-20) = 15 \times 5 = 75$ — five units above baseline, each worth $15$.

2. Age and distance's contributions

   • Age: $w_{\text{age}}(5-15) = -2\times(-10) = 20$ — younger than baseline, and age's weight is negative, so a below-baseline age adds value.
   • Distance: $-5\times(8-10) = 10$, the same logic.
3. Renovated's contribution, and the total

   Renovated: $20\times(1-0)=20$. All four sum to $75+20+10+20=125$, and $320 + 125 = 445$ — exactly the predicted price, with no approximation anywhere.