Sobol

Sobol indices for the Orthogonal Additive Kernel.

Computes analytic Sobol indices per interaction order for the constrained SE kernel under standard normal input density.

Reference

Lu, X., Boukouvalas, A., & Hensman, J. (2022). Additive Gaussian Processes Revisited. ICML. (Eq. 14, Appendix G.1)

sobol_indices

sobol_indices(
    kernel: OrthogonalAdditiveKernel,
    x_train: Float[Array, "N D"],
    y_train: Float[Array, "N 1"],
    noise_variance: float,
) -> Float[Array, " D_tilde"]

Compute normalized Sobol indices per interaction order.

The Sobol index for interaction order d measures what fraction of the posterior variance is explained by d-th order interactions. Indices are normalized to sum to 1.

Uses vmap for per-dimension integral matrices and matrix-level Newton-Girard to sum over all subsets of each order (no Python loops in the computation path).

Parameters:

• kernel (OrthogonalAdditiveKernel) –

  A fitted OrthogonalAdditiveKernel.

• x_train (Float[Array, 'N D']) –

  Training inputs of shape (N, D).

• y_train (Float[Array, 'N 1']) –

  Training targets of shape (N, 1).

• noise_variance (float) –

  Observation noise variance.

Returns:

• Float[Array, ' D_tilde'] –

  Array of shape (max_order,) with normalized Sobol indices

• Float[Array, ' D_tilde'] –

  for orders 1 through max_order.