On adaptive inference and confidence bands

By Marc Hoffmann, Richard Nickl. 2011

84 citations recorded by the source corpus.

The problem of existence of adaptive confidence bands for an unknown density $f$ that belongs to a nested scale of H\"{o}lder classes over $\mathbb{R}$ or $[0,1]$ is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of H\"{o}lder balls $\Sigma(r),\Sigma(s),r\ne s$, of fixed radius, a nonparametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and sufficient for the existence of honest asymptotic confidence bands, and that it is strictly weaker than similar analytic conditions recently employed in Gin\'{e} and Nickl [Ann. Statist. 38 (2010) 1122--1170]. The exceptional sets for which honest inference is not possible have vanishingly small probability under natural priors on H\"{o}lder balls $\Sigma(s)$. If no upper bound for the radius of the H\"{o}lder balls is known, a price for adaptation has to be paid, and near-optimal adaptation is possible for standard procedures. The implications of these findings for a general theory of adaptive inference are discussed.

Subfields: Computer Vision and Pattern Recognition, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9b38bf64a34519439edff4f90fe86b0947e64365

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Citations in the corpus

Anti-concentration and honest, adaptive confidence bands (2013, unknown): “For example, if we are interested in the function class Σ(t, L) ∪ Σ(t′, L) for some t′ > t, [24] showed that it is necessary and sufficient to exclude functions Σ(t, L)\Σ(t, L, ρn) where Σ(t, L, ρn) = {f ∈”
Adaptive confidence sets in $$L^2$$ (2011, result): “The occurrence of this rate in Theorem 1 parallels similar findings in Theorem 2 in [17] in the different situation of confidence bands, and is inspired by the general ideas in [5,13,17,22], which attempt to find ‘maximal’ subsets of the usual parameter spaces of adaptive estimation for which honest”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”
Confidence sets in sparse regression (2012, methodology): “The main idea, which is the same for all theorems, follows Hoffmann and Nickl (2011), Bull and Nickl (2013) to solve the composite testing problem H0 : θ ∈B0(k0) vs.”
Honest adaptive confidence bands and self-similar functions (2011, methodology): “Different conditions have been considered by Picard and Tribouley (2000), Genovese and Wasserman (2008), Giné and Nickl (2010), and Hoffmann and Nickl (2011).”
On adaptive posterior concentration rates (2013, methodology): “(36) In this case, there exists no adaptive confidence band, see for instance Hoffmann and Nickl [19], so that (36) implies that for all α > 0, lim n inf β1≤β≤β2 inf θ0∈H(β,L) Pnθ0(θ0 ∈ Cn) = 0.”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, background): “functionals, [7, 14] for adaptive confidence bands and [10, 22] for adaptive confidence balls.”