SHAP：一次預測怎麼分配到各個特徵？

這次模型輸出 = 基準值 + 特徵1的歸因 + … + 特徵M的歸因
f(x*)        = phi0  + phi1          + … + phiM

p = 1 / (1 + exp(−[phi0 + sum(phi_i)]))

f(x,z) = x + 2z + 3xz
待解釋點 (x,z) = (1,1)，f(1,1) = 6

第一種賽局：只有一個背景點 (0,0)
已知的欄位固定在目標值1；未知欄位用背景值0

phi_i = sum over S subset of F without i:
          |S|! × (M−|S|−1)! / M!
          × [v(S union {i}) − v(S)]

M：玩家數／特徵數
權重：在所有等權排列裡，恰好由S先於i出現的比例

背景：(0,0)、(1,1)，每點機率1/2

phi0 = b + sum_j w_j E[X_j]
phi_j = w_j × (x*_j − E[X_j])

v_cond(S) = E[f(X) | X_S = x*_S]

python3 two_feature_shapley.py

"""Exact two-feature attribution games; no SHAP library or model fitting.

Marginal replacement and observational conditioning define different games.
Neither operation is a causal intervention in this example.
"""

from fractions import Fraction as F
from itertools import permutations
import json


BACKGROUND = ((0, 0), (1, 1))
TARGET = (1, 1)


def value(model, coalition, mode, background):
    if mode == "conditional":
        matching = [row for row in background
                    if all(row[j] == TARGET[j] for j in coalition)]
        if not matching:
            raise ValueError("No observed support for this conditioning event")
        return sum((F(model(*row)) for row in matching), F(0)) / len(matching)
    if mode != "marginal":
        raise ValueError("Unknown game")
    predictions = []
    for row in background:
        combined = tuple(TARGET[j] if j in coalition else row[j] for j in (0, 1))
        predictions.append(F(model(*combined)))
    return sum(predictions, F(0)) / len(predictions)


def explain(name, model, mode, background):
    coalitions = (frozenset(), frozenset({0}), frozenset({1}), frozenset({0, 1}))
    game = {s: value(model, s, mode, background) for s in coalitions}
    contributions = [F(0), F(0)]
    orders = []
    for order in permutations((0, 1)):
        known = frozenset()
        increments = []
        for j in order:
            expanded = known | {j}
            increment = game[expanded] - game[known]
            contributions[j] += increment / 2
            increments.append(str(increment))
            known = expanded
        orders.append({"order": list(order), "increments_in_order": increments})
    base = game[frozenset()]
    prediction = F(model(*TARGET))
    assert base + sum(contributions, F(0)) == prediction
    return {
        "model": name, "game": mode, "background_rows_equal_mass": list(background),
        "target": TARGET,
        "coalition_values": {"empty" if not s else "+".join(str(j) for j in sorted(s)): str(v)
                             for s, v in game.items()},
        "orders": orders, "base": str(base), "phi": [str(v) for v in contributions],
        "prediction": str(prediction), "base_plus_phi_exact": str(base + sum(contributions)),
    }


def main():
    interaction = lambda x, z: x + 2*z + 3*x*z
    reads_x_only = lambda x, z: x
    cases = [
        explain("x+2z+3xz", interaction, "marginal", ((0, 0),)),
        explain("x+2z+3xz", interaction, "marginal", BACKGROUND),
        explain("x+2z+3xz", interaction, "conditional", BACKGROUND),
        explain("x (z unused by model)", reads_x_only, "marginal", BACKGROUND),
        explain("x (z unused by model)", reads_x_only, "conditional", BACKGROUND),
    ]
    print(json.dumps({
        "scope": "Exact enumeration of hand-specified two-feature games; no fitted explainer, uncertainty estimate, or causal effect",
        "feature_indices": {"0": "x", "1": "z"},
        "conditional_scope": "Only explicit finite background support; conditioning events used here have positive mass",
        "cases": cases,
    }, ensure_ascii=False, indent=2))


if __name__ == "__main__":
    main()

The MIT License (MIT)

Copyright (c) 2018 Scott Lundberg

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# DS57 source image and license notice

These illustrations reproduce four distinct saved notebook documentation examples. The source repository grant covers its associated documentation; the Adult dataset has a separate grant and provenance. No dataset is included.

## Source plots

**Source:** Scott Lundberg, SHAP project documentation at pinned commit [`f238480bb9dddf663d2bbc161a3d0b3c282e0386`](https://github.com/shap/shap/tree/f238480bb9dddf663d2bbc161a3d0b3c282e0386), under the repository [MIT license](https://github.com/shap/shap/blob/f238480bb9dddf663d2bbc161a3d0b3c282e0386/LICENSE). The repository license includes associated documentation. The complete copyright notice and terms are reproduced in the article’s SHAP MIT license section.

- `images/fig-01-waterfall.png` — [waterfall notebook, cell 3](https://github.com/shap/shap/blob/f238480bb9dddf663d2bbc161a3d0b3c282e0386/notebooks/api_examples/plots/waterfall.ipynb). The complete output PNG is copied byte-for-byte. It shows one sampled row, with source-reported XGBoost margin/log-odds, expected value and contributions. No mobile crop is provided because splitting the complete cumulative path would separate row contributions from the base and endpoint; the complete original plot is retained.
- `images/fig-02-beeswarm.png` — [beeswarm notebook, cell 3](https://github.com/shap/shap/blob/f238480bb9dddf663d2bbc161a3d0b3c282e0386/notebooks/api_examples/plots/beeswarm.ipynb). Complete source PNG copied byte-for-byte. `images/fig-02-beeswarm-mobile.png` places the complete plot first, followed by two overlapping source-pixel x-range crops. Feature rows and original plotted points are retained; the whole image provides the full scale. The original high/low colorbar encodes feature values, not SHAP direction.
- `images/fig-03-heatmap.png` — [heatmap notebook, cell 3](https://github.com/shap/shap/blob/f238480bb9dddf663d2bbc161a3d0b3c282e0386/notebooks/api_examples/plots/heatmap.ipynb). Complete source PNG copied byte-for-byte. `images/fig-03-heatmap-mobile.png` places the complete plot first, then three overlapping source-column crops across its full height. The source rows, colors, curve and bars are unmodified. Columns are instances clustered by explanation similarity, not time; the whole image preserves the full row/scale context.
- `images/fig-04-decision.png` — [decision plot notebook, cell 8](https://github.com/shap/shap/blob/f238480bb9dddf663d2bbc161a3d0b3c282e0386/notebooks/api_examples/plots/decision_plot.ipynb). Complete source PNG copied byte-for-byte. No mobile crop is provided because the full feature order and cumulative path are needed together; the complete original image is retained.
- `images/cover-ds57-waterfall.png` — the complete original cell-3 waterfall plot proportionally fitted and centered on a white 1600×900 canvas. No title, text or graphic marks are added.

The two mobile layouts are adaptations: they stack the unmodified complete source image first, then source-pixel crops on white. No labels, arrows, points, colors, values or axes are drawn or changed. Crop coordinates are recorded in `source-crop-audit.json`.

## Data attribution and limits

The SHAP notebooks use a GitHub-hosted processed copy of the UCI Adult data. The upstream UCI page credits Barry Becker and Ronny Kohavi (1996), [DOI 10.24432/C5XW20](https://doi.org/10.24432/C5XW20), and provides a separate [CC BY 4.0 grant](https://creativecommons.org/licenses/by/4.0/). Exact row/byte identity between the repository helper copy and the current UCI distribution has not been verified. No dataset is redistributed; the data license does not replace the plot authors’ MIT attribution.

These plots come from different notebook runs: waterfall uses a 2,000-row sample and XGBoost margin output; beeswarm uses its own XGBoost fit; heatmap uses another XGBoost configuration and the first 1,000 explained instances; decision plot uses a separate LightGBM split and its first 20 test rows. The decision notebook also uses that split as a validation set for early stopping. Do not present the four images as a matched experiment, untouched-test evaluation, or a reproduction by this asset build.

Original Lundberg and Lee NeurIPS 2017 paper figure pixels are not used because positive republication permission for those images was not established; the paper and its official errata remain link-only.