GP density estimation contraction rate via the concentration function

The density is f = exp(psi) / int exp(psi) with psi a Gaussian process, which forces positivity and normalisation automatically – the reason for the exponential link rather than a prior on f itself. The rate then follows from the concentration function, and the KERNEL decides the answer: a Matern process with smoothness matched to psi0 attains the minimax n^(-s/(2s+1)), while a squared-exponential process has analytic sample paths, is far too smooth for a merely s-smooth truth, and contracts only LOGARITHMICALLY unless its length scale is itself given a prior. That contrast is returned rather than buried.

Usage

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

Arguments

x

Observations; used for the sample size.

s

Smoothness of the log-density; 1 when NULL.

n

Sample size; taken from x when NULL.

kernel

One of "squared_exponential", "matern", "rescaled_se".

Value

List with n, smoothness, kernel, rate, minimax_rate, attains_minimax, rate_kind, ratio_to_minimax, link, driver, method.

Details

Formula: minimax = n^(-s/(2s+1)); rescaled SE picks up (log n)^((s+1)/(2s+1)); plain SE gives (log n)^(-s).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 11.3.1 and ch. 11; van der Vaart & van Zanten.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosalgpdenscrt(V)
#> $n
#> [1] 8
#> 
#> $smoothness
#> [1] 1
#> 
#> $kernel
#> [1] "squared_exponential"
#> 
#> $rate
#> [1] 0.4808983
#> 
#> $minimax_rate
#> [1] 0.5
#> 
#> $attains_minimax
#> [1] FALSE
#> 
#> $rate_kind
#> [1] "LOGARITHMIC"
#> 
#> $ratio_to_minimax
#> [1] 0.9617967
#> 
#> $link
#> [1] "f = exp(psi) / int exp(psi): positivity and normalisation for free"
#> 
#> $driver
#> [1] "the concentration function: RKHS approximation + small-ball probability"
#> 
#> $method
#> [1] "GP density contraction (Sec. 11.3.1); the kernel's smoothness decides the rate"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"