Adaptive Polya tree rate

\(\varepsilon_n \asymp n^{-s/(2s+1)}\log n\) for an s-smooth \(p_0\) (Sec. 7.2.3). The prior \(a_m = m^2\) attains this for every s WITHOUT being told s; the logarithm is exactly the price of that adaptation, and is returned separately so "near-optimal" is a number. Mirrors morie.fn.gh_pt_adapt.

Usage

morie_polya_tree_rate(x, s = NULL, n = NULL, levels = 6L, a_scale = 1)

Arguments

x

numeric observations, for the sample size.

s

smoothness to report at.

n

sample size to evaluate at.

levels, a_scale

recorded for the corresponding fit.

Value

list: n, smoothness, rate, minimax_rate, log_factor, ratio_to_minimax, adaptive, requires_knowing_s, scan, method.

References

Ghosal and van der Vaart, Sec. 7.2.3; Ch. 10.

Examples

morie_polya_tree_rate(numeric(5), s = 2, n = 10000)$ratio_to_minimax
#> [1] 9.21034