Gaussian process RKHS, Euclidean case

For W ~ N(0, Sigma) on R^k the reproducing-kernel Hilbert space is the range of Sigma, carrying the inner product <Sigma a, Sigma b>_H = a' Sigma b. This finite-dimensional case is the one where every object in the general definition can be written down, which is why the book uses it to fix intuition; the reproducing formula h(t) = <h, K(t, .)>_H is checked coordinatewise here.

Usage

Ghosalgpdefrkhs(Sigma, a, b)

Arguments

Sigma

Covariance matrix, k by k.

a, b

Coefficient vectors of length k.

Value

List with estimate (the inner product), h, reproducing_gap, method.

Details

Formula: <Sigma a, Sigma b>_H = a' Sigma b; h = Sigma a; K(t, .) = Sigma e_t, so <h, K(t, .)>_H = a' Sigma e_t = h(t).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, Example 11.15 and Definition 11.12, eq. (11.8).

Examples

S <- matrix(c(2, 0.5, 0.5, 1), 2)
Ghosalgpdefrkhs(S, a = c(1, 0), b = c(0, 1))$estimate   # = a' Sigma b = 0.5
#> [1] 0.5