Stein Kernel Loss

Use this skill when a task needs to evaluate an empirical KSD or scalar diffusion KSD objective for a one-dimensional unnormalized model. It is appropriate for bounded recovery experiments, diagnostics, and cross-checks of minimum Stein discrepancy estimators. Do not use it as a general multidimensional matrix-kernel implementation.

Inputs

• samples: at least two numeric observations.
• theta: model parameter used by the score and diffusion functions.
• score_fn(x, theta): derivative of the log unnormalized density with respect to x.
• bandwidth: positive Gaussian-kernel bandwidth.
• Optional diffusion_fn(x, theta) and diffusion_derivative_fn(x, theta) for scalar DKSD.

Outputs

• Pairwise scalar Stein-kernel values.
• U-statistic loss over all ordered distinct pairs.
• Diagnostics containing pair count and finite-value status.

Workflow

1. Define the model score without using normalizing constants.
2. Choose identity diffusion for vanilla KSD or a scalar diffusion and derivative for DKSD.
3. Call u_statistic_loss from scripts/stein_loss.py for a candidate parameter.
4. Repeat over candidate parameters or pass the objective to an optimizer skill.
5. Save diagnostics when the value is used as recovery evidence.

Validation

Run python -m pytest tests or validate through the Distiller skill-tree validator with --run-tests.

Limitations

This skill implements the scalar one-dimensional special case used for fast recovery. It intentionally does not implement the full matrix-valued RKHS formula or Riemannian geometry from the paper.