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Conservative coverage of adaptive empirical-Bayes credible balls on polished tails

Statement

Use the Gaussian sequence procedure of Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model with fixed $p \geq 0$ , $C \geq 1$ , $A > 0$ , and $C^{- 2} i^{- 2 p} \leq κ_{i}^{2} \leq C^{2} i^{- 2 p}$ . Fix $L_{0} \geq 1$ , $N_{0} \geq 1$ , $ρ \geq 2$ , and $γ \in (0, 1)$ . Assume that $θ_{0} \in Θ_{p t} (L_{0}, N_{0}, ρ)$ and $θ_{0, 1} = 0$ . Then there is a finite inflation constant $L > 0$ , depending only on the fixed model and class parameters, such that $\inf_{\begin{array}{c} θ_{0} \in Θ_{p t} (L_{0}, N_{0}, ρ) \\ θ_{0, 1} = 0 \end{array}} P_{θ_{0}}^{(n)} {θ_{0} \in {\hat{C}}_{n} (L)} ⟶ 1.$ This is conservative frequentist coverage tending to one, not exact asymptotic coverage $1 - γ$ . The source also allows a sequence of caps $A_{n} > 0$ satisfying, for all sufficiently large $n$ , $A_{n} \leq \frac{\sqrt{\log n}}{4 \sqrt{\log (ρ \lor e)}} .$ In that case there is a constant $K_{i n f l} > 0$ , depending only on the fixed model and class parameters and not on $n$ , such that the same coverage conclusion holds for any deterministic inflation sequence satisfying $L_{n} \geq K_{i n f l} (3 ρ^{3 (1 + 2 p)})^{A_{n}}$ for all sufficiently large $n$ .

Facts & Assumptions

Given: AC and the hypotheses in the Statement, with either fixed $A$ or the stated growing-cap regime.

[F1]

The empirical-Bayes procedure defines its credible set to be all of $ℓ^{2}$ when the selector is zero. Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model

[F2]

The polished-tail bracket places the selector in $[a_{n}, b_{n}] = [{\underline{α}}_{n}, {\overline{α}}_{n}]$ with probability tending uniformly to one and compares $R_{a_{n}} / R_{b_{n}}$ , where $R_{α} = n^{- 2 α / (1 + 2 α + 2 p)}$ . Marginal-likelihood maximizer bracket on polished tails

[F3]

For bracket endpoints with $b_{n} > 0$ , the posterior-radius, bias and noise bounds apply when $h_{n} (b_{n}; θ_{0}) \leq 8 C^{8}$ . Posterior radius, bias and noise comparisons for empirical Bayes

Proof

1.1F2F3givenalgebra

Write $a = a_{n}$ , $b = b_{n}$ . Continuity of $h_{n} (\cdot; θ_{0})$ gives $h_{n} (b; θ_{0}) \leq 8 C^{8}$ whenever $b > 0$ : if the sublevel set is nonempty, its supremum belongs to it by compactness, while an empty sublevel set has $b = 0$ by convention. The hypotheses $N_{n, A_{n}} \to \infty$ and $A_{n} / N_{n, A_{n}} \to 0$ in [F3] hold for fixed $A$ and under the growing cap, because $\log N_{n, A_{n}} = \log n / (1 + 2 A_{n} + 2 p)$ grows at least as a positive multiple of $\sqrt{\log n}$ when $A_{n}$ is unbounded.

2.1F1F2F3step 1.1algebra

On the bracket event from [F2], the case ${\hat{α}}_{n} = 0$ gives coverage by [F1]. Otherwise $b > 0$ . Couple the centered posterior-mean noise by the same normal errors for every $α$ . Each posterior variance decreases with $α$ , so its norm quantile decreases too, and [F3] gives $r ({\hat{α}}_{n}) \geq r (b) \geq \sqrt{d R_{b}}$ . The triangle inequality and [F3] then give, with probability tending uniformly to one, $∥ {\hat{θ}}_{n, {\hat{α}}_{n}} - θ_{0} ∥ \leq K_{1} (1 + ρ^{1 + 2 b + 2 p}) \sqrt{R_{b}} + K_{2} \sqrt{R_{a}}$ . The probability error for the noise is bounded by $K / N_{n, A_{n}}$ and is independent of the truth.

3.1F2step 2.1choosealgebra

For fixed $A$ , [F2] bounds $R_{a} / R_{b}$ by a constant depending only on the fixed parameters. Choose $L$ so that $L \sqrt{d}$ exceeds $K_{1} (1 + ρ^{1 + 2 A + 2 p}) + K_{2} \sqrt{R_{a} / R_{b}}$ uniformly. Then step 2.1 implies coverage on an event whose probability tends uniformly to one. This proves the fixed-cap conclusion.

4.1F2step 2.1step 3.1algebra∎

For growing $A_{n}$ , the sharpened comparison in [F2] bounds $R_{a} / R_{b}$ by $K (1 + L_{0} ρ^{1 + 2 A_{n}})^{1 + 2 p}$ , with $K$ independent of $n$ and the truth. Hence $\sqrt{R_{a} / R_{b}} \leq K_{n o i s e} ρ^{(1 + 2 p) A_{n}}$ after absorbing fixed factors into $K_{n o i s e}$ . The bias bound from step 2.1 is at most $K_{b i a s} ρ^{2 A_{n}} \sqrt{R_{b}}$ , with $K_{b i a s}$ also independent of $n$ and the truth. Since $(3 ρ^{3 (1 + 2 p)})^{A_{n}}$ dominates both $ρ^{(1 + 2 p) A_{n}}$ and $ρ^{2 A_{n}}$ , choose a fixed $K_{i n f l} \geq 1$ with $K_{i n f l} \sqrt{d} \geq K_{n o i s e} + K_{b i a s}$ . Then the bound on $L_{n}$ makes the credible radius at least the sum of the bias and noise bounds on the bracket and noise events. The bracket and noise errors tend uniformly to zero because $N_{n, A_{n}} \to \infty$ , proving the growing-cap conclusion.

Remarks

The first-coordinate restriction is necessary for uniformity with this prior convention: the marginal likelihood does not use its value in selecting $α$ , while its posterior bias contains that value. All probability-model choice assumptions are inherited from the stated procedure.

Depends on

Used by

A power-law truth lies in the polished-tail class Example

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Botond Szabó, Aad van der Vaart and Harry van Zanten, Frequentist coverage of adaptive nonparametric Bayesian credible sets, Annals of Statistics 43(4) (2015) (standard reference, not scraped)
B. T. Knapik, A. W. van der Vaart and J. H. van Zanten, Bayesian inverse problems with Gaussian priors, Annals of Statistics 39(5) (2011) (standard reference, not scraped)