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name: stein_kernel_scoring description: Compute Kernelized Stein Discrepancy pairwise Stein kernels and U/V-statistic estimates from samples, model scores, and an RBF kernel.

Stein Kernel Scoring

Use this skill when implementing a goodness-of-fit or sample-quality check from Liu, Lee, and Jordan's Kernelized Stein Discrepancy paper. It is appropriate when samples are available from an unknown distribution and the candidate model can provide score values ∇ log q(x), but likelihood normalizers should not be evaluated.

Do not use this skill for two-sample MMD tests, likelihood-ratio tests, or cases where no differentiable score function is available.

Inputs

• samples: numeric array-like data with shape (n, d) or one-dimensional shape (n,).
• scores: score vectors for the same samples, or a Python callable that returns them.
• bandwidth: positive RBF bandwidth, or None to use median pairwise distance.

Outputs

• Pairwise Stein kernel matrix for the RBF kernel.
• Unbiased U-statistic KSD estimate, excluding diagonal entries.
• V-statistic KSD diagnostic, including diagonal entries.
• Diagnostics for finite values, symmetry, and bandwidth.

Workflow

1. Convert samples and scores to two-dimensional numeric arrays.
2. Select a positive RBF bandwidth; median-distance selection falls back to 1.0 for degenerate samples.
3. Compute the RBF Stein kernel k * [s_x·s_y + s_x·(x-y)/h² + (y-x)·s_y/h² + d/h² - ||x-y||²/h⁴].
4. Compute U-statistic and V-statistic estimates from the pairwise matrix.
5. Report diagnostics rather than silently correcting invalid scores.

Validation

Run:

python tests/test_stein_kernel_scoring.py

The tests check shape, symmetry, diagonal handling, and stronger KSD under a shifted model score.

Limitations

• The deterministic script implements the RBF kernel only.
• The score function must already be differentiable and correctly evaluated by the caller.
• Median bandwidth is a practical experiment heuristic, not a theorem requirement.