The observation x is a complete input row, with every feature value known. A fitted model f predicts its outcome: ŷ(x) = f(x). The observed outcome y is used in fitting; the explanation concerns the model’s prediction. It does not compare that prediction with an observed outcome.

The feature universe F has m members. The preceding group S contains features already fixed to x; feature i is the additional feature, outside S. The value v_x(S) is the average of the fixed model’s predictions over hybrid rows. Included columns take their values from x; excluded columns take theirs from the background. This averaging is what we mean by integrating out the excluded features. Every model call still uses every input column.

With nothing fixed, v_x(∅) is the background average prediction. With everything fixed, every hybrid row equals x, so v_x(F) = ŷ(x). Adding no feature leaves a group unchanged: S ∪ ∅ = S.

With the model and background held fixed, ϕ_i(x) is shorthand for ϕ_i(v_x). It gives feature i’s credit relative to the background average, in prediction units. Repeat the calculation for each feature to recover the prediction as baseline plus contributions.

The Bread and Peace warm-up uses bread, peace, and observed incumbent-party two-party vote share: each has mean zero and SD one. Bread is standardized real income growth. Peace is minus standardized war fatalities, so higher peace means fewer fatalities. The stipulated correlations are bread–peace ρ = +0.5, bread–vote +0.75, and peace–vote +0.75. In a standardized univariate regression, the slope equals the feature–outcome correlation, giving bread-only and peace-only slopes of +0.75.

For each bivariate slope, subtract the feature correlation times the other feature’s outcome correlation, then divide by 1 − ρ²:

The fitted bivariate equation is ŷ(x) = 0.5 × bread + 0.5 × peace. Centering makes all three intercepts zero. Predictions use the outcome’s SD units; they are not themselves rescaled to SD one. A peace value of +1 means one SD fewer war fatalities than average.

Refitting without peace produces 0.75 = 0.5 + (0.5)(0.5): the full-model bread coefficient plus the omitted-variable term. SHAP keeps the bivariate coefficients fixed and changes which inputs are averaged. With this additive model and overall background means of zero, a feature’s marginal is the same in either order, despite the correlation between the features.

This is a teaching example inspired by Douglas Hibbs’s Bread and Peace model. The correlations and election profiles are constructed, rather than estimated from historical elections. Hibbs’s original specification also estimates income-growth time weights; this warm-up uses direct two-feature OLS and reverses the standardized war-fatalities measure to define peace.

Call the fixed prediction rule f, so f(x) = ŷ(x). In a machine-learning application, training inputs X and outcomes y are used to fit it once. Here we supply a transparent earnings equation so every prediction can be checked. The explanation procedure holds that equation fixed:

The three features are ability, neighborhood opportunity (written as neighborhood), and experience. Ability and experience range from 0 to 1; neighborhood opportunity ranges from −1 to +1. The interaction ability × neighborhood is computed inside f, rather than treated as a fourth player. It can outweigh ability’s positive main term in an adverse neighborhood.

The eight equally weighted background rows cover every endpoint combination: low or high ability, adverse or favorable neighborhood, and low or high experience. Their average prediction is $47,000. They define our reference distribution; their observed outcomes are not needed for the calculation. Zero is a legitimate input value, rather than a marker for an excluded feature.

Here the adverse context makes revealing this person’s ability lower the averaged prediction. This sign reversal comes from the interaction in our equation.

The equation and reference people are constructed for teaching. SHAP assigns credit for its prediction; it does not estimate a causal effect of changing a person’s features. Choosing another person changes the coalition values and credits while the equation, background, and baseline stay fixed.

For three features, there are 3! = 6 equally weighted revealing orders. A particular preceding group S appears in |S|!(3 − |S| − 1)! of them: arrange the features in S before feature i, and the remaining features after it. Divide this count by six to get its weight.

The empty group gets 0!2!/3! = 2/6; each one-feature group gets 1!1!/3! = 1/6; the two-feature group gets 2!0!/3! = 2/6. These weights sum to one across the four preceding groups that exclude i.

In the earnings example, ability’s marginal depends on whether neighborhood is already fixed, and neighborhood’s depends on whether ability is already fixed. Experience’s position does not affect those marginals because experience enters additively. Each of the two contexts has total weight 3/6. Experience itself has the same marginal in all six orders, giving total weight one.

Removing peace and refitting OLS changes the bread coefficient. The omitted-variable identity compares that new coefficient with the full-model coefficient. Our SHAP calculation instead keeps the full fitted model and averages over the background peace distribution.

Conditional SHAP is another possible game: it averages excluded features using their distribution conditional on the included values. Correlated features can then share credit differently, even in a linear model. That still keeps the prediction model fixed; it is not refitting a reduced model. See SHAP’s linear-model explanation.

In the linear OLS warm-up, averaging predictions equals predicting with excluded features at their overall background means. The row-averaging definition also works for nonlinear and interaction models, where substituting a single mean row need not reproduce the average prediction.

The transparent equation lets us check the arithmetic. In a machine-learning application, the prediction function below can be replaced by an already fitted model’s predict method. The model remains fixed while SHAP explains its output.

import numpy as np
import shap

def predict(rows):
    ability, neighborhood, experience = rows.T
    return 40 + 10*ability + 24*ability*neighborhood + 4*experience

def coalition_value(revealed):
    hybrid = background.copy()
    hybrid[:, revealed] = x[revealed]
    return predict(hybrid).mean()

# Ability joins after neighborhood.
before = coalition_value([1])
after = coalition_value([1, 0])
marginal = after - before

masker = shap.maskers.Independent(
    background, max_samples=len(background)
)
explainer = shap.ExactExplainer(predict, masker)
explanation = explainer(x.reshape(1, -1))
shap.plots.waterfall(explanation[0])

Here background holds the eight reference rows and x is the selected person’s three inputs. The reproduction files define those arrays, enumerate every coalition, and verify the values against Python SHAP. Larger models use explainers suited to their model class and computational budget.