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myst:
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    description: "Understand prediction neutrality, empirical reference distributions, and localization in prediction space."
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# 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](references.md).

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](references.md) develop the reference-distribution viewpoint
for Integrated Gradients and Shapley attribution respectively.