K(x, y) = sum_k lambda_k phi_k(x) phi_k(y). Mercer's theorem says
every continuous kernel has such an expansion, and the random series
W = sum_k a_k(t) Z_k with independent standard normal Z_k has exactly
that covariance – so a series prior and a Gaussian process prior are
the same object viewed two ways, which is what makes the RKHS
computations of the chapter tractable. Cosine eigenbasis with
lambda_k = k^(-2) here; the partial sums converge.
Usage
Ghosalseriesgp(x = 0.3, y = 0.7, n_terms = 60)
Arguments
- x, y
Evaluation points.
- n_terms
Number of terms summed; at least 3 so the convergence
check has three partial sums to compare.
Value
List with estimate (the kernel value),
partial_sums, converging, method.
Details
Formula: K(x, y) = sum_k k^(-2) phi_k(x) phi_k(y),
phi_k(t) = sqrt(2) cos(k pi t).
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, Examples 11.4 and 11.16.
Examples
Ghosalseriesgp()
#> $estimate
#> [1] -0.7564009
#>
#> $partial_sums
#> [1] -0.7624256 -0.7544100 -0.7564009
#>
#> $converging
#> [1] TRUE
#>
#> $method
#> [1] "eigenexpansion GP kernel (GvdV 2017 Ex 11.16)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
