模型解釋在解釋什麼？全局、局部與因果的差別

PDP(a) = (1/n) × sum_i f(a, z_i)

對每個想畫在橫軸的 a：
  複製參考資料 → 全部列的 x 改為 a → 模型預測 → 平均

第 i 列的 ICE_i(a) = f(a, z_i)

f(x,z) = x + 2z + 3xz

固定 z=0：ICE_0(x) = x
固定 z=1：ICE_1(x) = 2 + 4x

PDP(x) = 0.5 × x + 0.5 × (2+4x)
       = 1 + 2.5x

若 score 是越大越好：
importance_j = baseline_score − permuted_score_j

系統 A：X = U，Y = U
系統 B：X = U，Y = X

python3 explanation_scope_toy.py

"""Small deterministic examples for global, local, and causal questions.

This script does not fit a model or estimate an effect from real data.
"""

import json


def f(x, z):
    return x + 2 * z + 3 * x * z


def mse(actual, predicted):
    return sum((a - p) ** 2 for a, p in zip(actual, predicted)) / len(actual)


def main():
    # Observed support contains only (x, z) = (0, 0) and (1, 1), equally often.
    observed = [(0, 0), (0, 0), (1, 1), (1, 1)]
    pdp = {}
    ice = {}
    for x in (0, 1):
        by_z = {z: f(x, z) for z in (0, 1)}
        pdp[str(x)] = sum(by_z.values()) / 2
    for z in (0, 1):
        ice[str(z)] = {str(x): f(x, z) for x in (0, 1)}

    # Exact duplicate predictors: the fitted toy rule relies on x1 only.
    x1 = [0, 0, 1, 1]
    x2 = [0, 0, 1, 1]
    y = [0, 0, 1, 1]
    baseline_pred = list(x1)
    x1_shuffled = [1, 1, 0, 0]
    x2_shuffled = [1, 0, 0, 1]
    perm_x1_pred = list(x1_shuffled)
    perm_x2_pred = list(x1)  # The fixed model ignores x2.

    # Paired deterministic SCMs. Both have observational U -> X and X == Y.
    # The intervention replaces X's structural equation and holds U fixed.
    u_values = (0, 1)
    observational = [(u, u) for u in u_values]
    scm_a_do = {str(x): sum(u for u in u_values) / 2 for x in (0, 1)}
    scm_b_do = {str(x): float(x) for x in (0, 1)}

    result = {
        "scope": "hand-specified deterministic arithmetic; no fitting, real data, or effect estimation",
        "global_vs_individual": {
            "model": "f(x,z)=x+2z+3xz",
            "observed_support_rows": [
                {"x": x, "z": z, "weight": 0.25, "f": f(x, z)}
                for x, z in observed
            ],
            "support_note": "Observed support has x=z only. PDP values at x!=z require model evaluation off observed support.",
            "pdp_equal_weight_z": pdp,
            "pdp_slope_from_x0_to_x1": pdp["1"] - pdp["0"],
            "ice_by_fixed_z": ice,
            "ice_slopes_from_x0_to_x1": {
                z: curve["1"] - curve["0"] for z, curve in ice.items()
            },
        },
        "single_fixed_permutation_example": {
            "rows": [
                {"x1": x1[i], "x2": x2[i], "y": y[i]}
                for i in range(len(y))
            ],
            "fixed_model": "prediction=x1",
            "metric": "mean squared error on these four rows",
            "baseline_mse": mse(y, baseline_pred),
            "one_fixed_x1_permutation": x1_shuffled,
            "x1_permuted_mse": mse(y, perm_x1_pred),
            "x1_permutation_score_drop": mse(y, perm_x1_pred) - mse(y, baseline_pred),
            "one_fixed_x2_permutation": x2_shuffled,
            "x2_permuted_mse": mse(y, perm_x2_pred),
            "x2_permutation_score_drop": mse(y, perm_x2_pred) - mse(y, baseline_pred),
            "scope_note": "One fixed shuffle for one fixed model and metric; not an uncertainty estimate, a universal feature value, or a causal effect. x2 duplicates x1 in these rows, but this fitted rule reads x1 only.",
        },
        "observationally_indistinguishable_scms": {
            "exogenous_support": [{"u": u, "probability": 0.5} for u in u_values],
            "scm_a": "X=U; Y=U (no structural dependence of Y on X)",
            "scm_b": "X=U; Y=X",
            "shared_observational_joint_xy": [
                {"x": x, "y": y_value, "probability": 0.5}
                for x, y_value in observational
            ],
            "mean_y_under_do_x": {
                "scm_a": scm_a_do,
                "scm_b": scm_b_do,
            },
            "do_effect_mean_y_x1_minus_x0": {
                "scm_a": scm_a_do["1"] - scm_a_do["0"],
                "scm_b": scm_b_do["1"] - scm_b_do["0"],
            },
            "scope_note": "These are two explicitly assumed structures with the same observational distribution and different intervention effects; they show non-identification from that distribution alone, not an effect learned from data.",
        },
    }
    assert result["global_vs_individual"]["pdp_slope_from_x0_to_x1"] == 2.5
    assert result["global_vs_individual"]["ice_slopes_from_x0_to_x1"] == {"0": 1, "1": 4}
    assert result["single_fixed_permutation_example"]["baseline_mse"] == 0
    assert result["single_fixed_permutation_example"]["x1_permuted_mse"] == 1
    assert result["single_fixed_permutation_example"]["x2_permuted_mse"] == 0
    assert result["observationally_indistinguishable_scms"]["do_effect_mean_y_x1_minus_x0"] == {
        "scm_a": 0.0,
        "scm_b": 1.0,
    }
    print(json.dumps(result, ensure_ascii=False, indent=2))


if __name__ == "__main__":
    main()

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Copyright (c) 2007-2026 The scikit-learn developers.
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  this software without specific prior written permission.

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CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

BSD 3-Clause License

Copyright (c) 2007-2024 The scikit-learn developers.
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.

* Redistributions in binary form must reproduce the above copyright notice,
  this list of conditions and the following disclaimer in the documentation
  and/or other materials provided with the distribution.

* Neither the name of the copyright holder nor the names of its
  contributors may be used to endorse or promote products derived from
  this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Source attribution for the model-explanation illustrations

The selected plots were generated in scikit-learn documentation. The scikit-learn FAQ permits repository images and images generated within the documentation under the BSD 3-Clause License, excluding the logo and separately credited third-party content.
FAQ: https://scikit-learn.org/stable/faq.html#how-can-i-obtain-permission-to-use-the-images-in-scikit-learn-for-my-work

Figures 01-04 and the centered-ICE cover use the captured 1.9.1 documentation/source records:
Copyright (c) 2007-2026 The scikit-learn developers.
License: https://github.com/scikit-learn/scikit-learn/blob/1.9.1/COPYING

Figure 05 uses the 1.8.0 documentation/source record:
Copyright (c) 2007-2024 The scikit-learn developers.
License: https://github.com/scikit-learn/scikit-learn/blob/1.8.0/COPYING

Both complete, version-specific BSD license copies accompany this notice. Source copyright years are preserved, not rewritten.

Figure sources:
https://scikit-learn.org/stable/auto_examples/inspection/plot_partial_dependence.html
https://scikit-learn.org/stable/auto_examples/inspection/plot_permutation_importance_multicollinear.html
https://scikit-learn.org/1.8/auto_examples/inspection/plot_causal_interpretation.html

Changes: Five main images are byte-identical copies of the selected original PNGs. Mobile Figure 01 shows the whole figure, then the original temperature and season panels enlarged. Mobile Figure 02 shows the whole figure, then the original temperature and humidity panels enlarged; the humidity panel retains its original shared-y-axis convention, without newly drawn numeric ticks. Mobile Figure 05 shows the whole synthetic-association pairplot first, followed by its complete first column enlarged, containing the degree KDE and four scatter rows with all original left y labels/ticks and bottom degree x labels/ticks; it repeats source pixels and is not a new causal graph. The cover proportionally fits the complete centered-ICE figure on a white 1600x900 canvas. No labels, curves, data or annotations were redrawn. These original documentation examples were not rerun for this article.

Separate dataset attribution:
- Bike Sharing: Hadi Fanaee-T (2013), UCI Machine Learning Repository, DOI 10.24432/C5W894, CC BY 4.0. https://archive.ics.uci.edu/dataset/275/bike+sharing+dataset
- The source example loads OpenML Bike_Sharing_Demand version 2, dataset 42712. Its metadata license field says Public; that field is not a CC BY statement. The upstream UCI page separately grants CC BY 4.0. Row-level/byte-level identity of the two releases was not verified. Season-label ordering has a version caveat; version 3 metadata must not be silently substituted for version 2.
- The OpenML description requests citation of Hadi Fanaee-T and Joao Gama, Event labeling combining ensemble detectors and background knowledge, DOI 10.1007/s13748-013-0040-3. Gama is co-author of this introductory paper, not listed as creator in the cited UCI dataset Creator field. This is dataset citation, not a claim to have reproduced that paper.
- Breast Cancer Wisconsin (Diagnostic): William Wolberg, Olvi Mangasarian, Nick Street, and W. Street (1993), UCI Machine Learning Repository, DOI 10.24432/C5DW2B, CC BY 4.0. https://archive.ics.uci.edu/dataset/17/breast-cancer-wisconsin-data-set
- Figure 05 uses synthetic variables generated by the documentation example, not either of the above datasets or real education/wage observations.

These attributions do not imply endorsement by the source authors or dataset creators. Plot licensing and dataset provenance remain separate.