Concentration-function terms

phi_{w0}(eps) = inf{ ||h||_H^2 / 2 : ||h - w0|| <= eps }

• log P(||W|| < eps). The two terms are the whole of Gaussian contraction theory: a decentering (RKHS approximation) term and a small-ball term, and the rate is where they balance. For a diagonal series process the infimum is attained by keeping the largest coordinates of w0 and zeroing the rest, so the first term is a finite sum; the small-ball probability is estimated on the l2 norm by Monte Carlo.

Usage

Ghosalrkhsnorm(f0_coefs, lambdas, eps, n_sim = 3000, seed = 42)

Arguments

f0_coefs

Coefficients of the target w0.

lambdas

Coordinate variances of the process, all positive.

eps

Radius.

n_sim

Monte Carlo draws for the small-ball term.

seed

Seed for the deterministic draws.

Value

List with estimate (phi), decentering_norm2, small_ball_exponent, method.

Details

Formula: ||h||_H^2 = sum_{i kept} f0_i^2 / lambda_i; small-ball exponent = -log P(sqrt(sum_i lambda_i Z_i^2) < eps).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, eq. (11.11).

Examples

Ghosalrkhsnorm(f0_coefs = c(1, 2, 3, 4, 5, 6, 7, 8), lambdas = c(1, 2, 3, 4, 5, 6, 7, 8), eps = 0.5)
#> $estimate
#> [1] 26.00637
#> 
#> $decentering_norm2
#> [1] 36
#> 
#> $small_ball_exponent
#> [1] 8.006368
#> 
#> $method
#> [1] "concentration function (GvdV 2017 eq. 11.11)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"