Explain one prediction three ways

Same house from Chapter 6: size $25$, age $5$, distance $8$, renovated. Switch lenses. Saliency reports the raw weight magnitudes — the same four numbers no matter which house you ask about. SHAP — short for SHapley Additive exPlanations — reports each feature's actual pull on this house's price, relative to a typical one. Counterfactual reports something else again: how much each feature would have to move to flip an approve/deny decision.

All three read the same four weights $w$, but ask three different questions:

• $\text{saliency}_i$ — feature $i$'s raw sensitivity: the magnitude of its weight, the same for every input.
• $w_i$ — the linear model's weight for feature $i$.
• $\text{SHAP}_i$ — feature $i$'s actual contribution to this specific house's prediction.
• $x_i$ — this house's actual value for feature $i$.
• $\bar x_i$ — feature $i$'s baseline (typical) value.
• $\Delta^{\text{cf}}$ — the counterfactual edit: how much each feature would need to change to flip the decision.
• $f(x)$ — the model's prediction for input $x$.
• $w$ — the full weight vector.

1. Saliency: sensitivity in general

   Saliency answers "how sensitive is the output to this feature, in general."

2. SHAP: this instance's actual pull

   SHAP answers "how much did this feature's actual value move this prediction away from typical."

3. Counterfactual: the cheapest edit

   Counterfactual answers "what's the cheapest edit that changes the decision."

4. Different questions, not competing answers

   None of them is wrong — they're just not the same question, and mixing them up is a real, common mistake.

Compare this house to the baseline house — a very different prediction, $445$ versus $320$. Saliency's bars don't move at all: they never looked at either house's actual values, only the model's fixed weights. SHAP's bars do move, because SHAP's entire job is measuring how far this instance sits from typical.

This house predicts $445$; call anything below a $500$ threshold "not yet approved" for this exercise:

1. Saliency says renovated matters most

   $|w| = (15, 2, 5, 20)$ for size, age, distance, renovated — renovated's raw sensitivity is the largest of the four, and that ranking would be identical for every house this model ever sees.

2. SHAP ranks size highest instead

   Size dominates here:

   • Size contributes $75$ to the SHAP value, and this house's size is $5$ units above baseline
   • Renovated contributes $20$ to the SHAP value, and this house's renovated status only differs by $1$
3. The counterfactual finds a different answer again

   This house's prediction is $445$, so $f(x) = 445-500 = -55$ below the threshold. The weight vector's squared norm is $\lVert w \rVert^2 = 15^2+(-2)^2+(-5)^2+20^2 = 225+4+25+400=654$, giving a scale factor of $-55/654\approx-0.0841$. Each feature's delta is $-\text{scale}\times w_i$:

   • Size: $0.0841\times15\approx+1.26$
   • Age: $-0.0841\times2\approx-0.17$
   • Distance: $-0.0841\times5\approx-0.42$
   • Renovated: $0.0841\times20\approx+1.68$

   That last number is a real problem: renovated is a $0/1$ feature, and $1.68$ isn't a valid value for it. The naive closed form doesn't know that "renovated" can't be anything other than $0$ or $1$.