Gaussian-process density contraction rate — morie_gp_density

The rate follows from the concentration function, whose two terms – an RKHS approximation term and a small-ball probability – are balanced at \(\varphi(\varepsilon_n) \le n\varepsilon_n^2\) (Sec. 11.3.1). The kernel decides the answer: Matern matched to the truth attains the minimax rate, while a SQUARED-EXPONENTIAL process has analytic sample paths and is far too smooth, contracting only logarithmically unless rescaled.

Usage

morie_gp_density_rate(x, s = NULL, n = NULL, kernel = "squared_exponential")

Arguments

x: numeric observations, for the sample size.
s: smoothness of the log-density.
n: sample size.
kernel: "squared_exponential", "matern" or "rescaled_se".

Value

list: n, smoothness, kernel, rate, minimax_rate, attains_minimax, ratio_to_minimax, rate_kind, link, driver, method.

Details

That contrast is asymptotic: at \(n = 10^4, s = 1\) the logarithmic rate is numerically SMALLER than the rescaled polynomial one. ratio_to_minimax is the honest summary – it diverges, the rate value alone does not. Mirrors morie.fn.gh_c11_4.

References

Ghosal and van der Vaart, Sec. 11.3.1 and Ch. 11.

Examples

morie_gp_density_rate(numeric(5), s = 1, n = 10000)$rate_kind
#> [1] "LOGARITHMIC"