Prediction-neutral backgrounds#

The prediction contrast#

A baseline distribution is part of the attribution question. For Integrated Gradients averaged over baselines \(b_i\) with normalized weights \(w_i\),

\[A_j(x)=\sum_i w_i\,\mathrm{IG}_j(x;b_i),\qquad \sum_j A_j(x)=f(x)-\sum_i w_i f(b_i).\]

For interventional Shapley attribution, the same reference mean is the empty-coalition value \(v(\emptyset)=\mathbb{E}_Q[f(X)]\). Both methods therefore decompose \(f(x)-f_0\) when the background mean equals \(f_0\). They need not assign that total to features in the same way.

If the achieved mean is \(f_0+r\), the attribution total is \(f(x)-f_0-r\), before any attribution numerical error. Report the residual instead of describing an approximate background as exactly neutral.

Localization and neutrality are different requirements#

The neutral set is \(\mathcal{M}_0=\{x:f(x)=f_0\}\). The canonical IG paper formulates the reference as the data distribution conditional on this set. Finite samples generally require a neighborhood around it rather than observations exactly on it. See Hentschel (2026a), Sections 2–3.

CBaseline localizes in prediction space, using a fitted metric that accounts for scale and redundant output directions. It uses observed feature rows without synthesizing new reference cases. High-dimensional features do not enter the localization metric, though prediction evaluation, storage, and attribution still depend on feature dimension. Many independent outputs can also make localization harder.

Calibration enforces a mean constraint; it does not make every retained row predict exactly f0. A reweighting that pairs high- and low-prediction cases averages to f0 while containing no case the model predicts anywhere near it. Two neutral distributions may explain the same total while allocating it differently. Localization identifies which reference population, among those with that mean, is relevant to the question.

What remains the attribution engine’s responsibility#

CBaseline does not compute attributions or change the fitted model. It provides the distribution for averaging IG paths or defining a SHAP background. Observed baseline endpoints do not imply that every point along an IG path, or every SHAP hybrid feature vector, is an observed input. Nor does a background choice turn a model explanation into a causal effect.

Keep one background fixed when comparing observations under the same question. Changing f0, the reference sample, or the background construction changes the reference population. Determinism is conditional on the fitted model, reference sample including row order, and construction settings; it does not remove randomness in model fitting or downstream approximation.

Comparison with common references#

Reference

Mean prediction

Observed rows

Localized near f0

Mean input

Generally differs from the mean output

Not necessarily

No

Full reference sample

Neutral at its own mean output

Yes

No

Random subsample

Matches the population mean only in expectation

Yes

Not generally

CBaseline calibrated

Matches feasible f0 to numerical tolerance

Yes

Yes, subject to widening

CBaseline equal-weight

Approximate; inspect residual

Yes

Yes, with a neutrality-directed slide

The two papers develop the reference-distribution viewpoint for Integrated Gradients and Shapley attribution respectively.