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name: ksd_bootstrap_gof description: Calibrate a KSD U-statistic goodness-of-fit test with the centered multinomial bootstrap for degenerate U-statistics.

KSD Bootstrap Goodness-of-fit Test

Use this skill when you already have a pairwise Stein kernel matrix and need the paper's KSD-U goodness-of-fit decision. The skill implements Algorithm 1's centered multinomial bootstrap, which is designed for the degenerate U-statistic null distribution.

Do not use this skill for ordinary non-degenerate bootstrap tests, V-statistic-only diagnostics, or linear-time KSD calibration.

Inputs

• stein_matrix: finite square matrix U where U[i,j] = u_q(x_i, x_j).
• alpha: significance level, usually 0.05.
• num_bootstrap: bounded number of bootstrap replicates.
• seed: integer random seed for reproducibility.

Outputs

• ksd_u: unbiased KSD U-statistic.
• scaled_observed: n * ksd_u.
• bootstrap_scaled: bootstrap null samples.
• p_value: fraction of bootstrap statistics greater than the observed statistic.
• reject: boolean decision p_value < alpha.

Workflow

1. Validate the square Stein matrix and exclude diagonal entries for observed KSD-U.
2. Draw multinomial bootstrap counts of size n with uniform category probabilities.
3. Center the counts as (w_i - 1/n) before forming pairwise products.
4. Compare the observed scaled statistic against the bootstrap scaled statistics using the strict > tail rule.
5. Log the seed and bootstrap count for reproducibility.

Validation

Run:

python tests/test_ksd_bootstrap_gof.py

The tests cover U-statistic calculation, reproducibility, and rejection behavior for a strongly positive Stein matrix.

Limitations

• Bootstrap counts are intended for bounded recovery and smoke checks; large paper-scale replicates may be expensive.
• The skill assumes the Stein matrix was computed correctly by a scoring skill.

Refinement Notes

Refinement invariant: treat multinomial draws as empirical probabilities before centering. For a draw count count_i, Equation (16) is implemented as (count_i / n - 1 / n), not raw count_i - 1 / n; raw counts make the bootstrap null too wide and suppress valid KSD-U rejections.