LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Posterior radius, bias and noise comparisons for empirical Bayes

Statement

Fix $γ \in (0, 1)$ . In the defined inverse Gaussian procedure, let $θ = (θ_{i}) \in ℓ^{2} (N_{+}, R)$ , where $N_{+} = {1, 2, \dots}$ , be the true sequence. For $n \geq 3$ and $α \in [0, A_{n}]$ write $q_{α} = 1 + 2 α + 2 p$ , $N_{n, α} = n^{1 / q_{α}}$ , $R_{n, α} = N_{n, α}^{- 2 α} = N_{n, α}^{1 + 2 p} / n$ , and $h_{n} (α; θ) : = \frac{q_{α}}{N_{n, α} \log n} \sum_{i = 1}^{\infty} \frac{n^{2} i^{1 + 2 α} (\log i) θ_{i}^{2}}{(i^{q_{α}} + n)^{2}} .$ Thus $h_{n} (α; θ)$ is exactly the score normalization $S_{α} / (N_{n, α} \log N_{n, α})$ in Uniform marginal-score bounds by derivative integration, since $\log N_{n, α} = (\log n) / q_{α}$ . Put $B_{i} (α) = - \frac{i^{1 + 2 α} θ_{i}}{i^{1 + 2 α} + n κ_{i}^{2}}, W_{i} (α) = \frac{\sqrt{n} κ_{i} Z_{i}}{i^{1 + 2 α} + n κ_{i}^{2}} .$ When $α$ is fixed in the proof, abbreviate $q = q_{α}$ , $N = N_{n, α}$ , and $R = R_{n, α}$ ; endpoint subscripts such as $q_{b}$ , $N_{a}$ , and $R_{b}$ denote the corresponding values. Let $0 \leq a \leq b \leq A_{n}$ , with $N_{n, A_{n}} \to \infty$ and $A_{n} / N_{n, A_{n}} \to 0$ . Constants below depend only on $p, C, L_{0}, ρ, γ$ . For all sufficiently large $n$ , (i) $r_{n, γ} (b)^{2} \geq d R_{n, b}$ for $b > 0$ ; (ii) if $θ_{1} = 0$ , $θ \in Θ_{p t} (L_{0}, N_{0}, ρ)$ and $h_{n} (b; θ) \leq 8 C^{8}$ , then $\sup_{a \leq α \leq b} ∥ B (α) ∥^{2} \leq K (1 + L_{0} ρ^{2 q_{b}}) R_{n, b}$ ; (iii) with probability at least $1 - K / N_{n, a}$ , $\sup_{a \leq α \leq b} ∥ W (α) ∥^{2} \leq K R_{n, a}$ . Part (iii) also holds at $a = 0$ . At $b = 0$ the procedure gives the whole parameter space whenever the selector lies in $[a, b]$ .

Facts & Assumptions

Given: AC and the hypotheses of the Statement, with independent standard normal errors $Z_{i}$ .

[F1]

The procedure and posterior formula give the independent Gaussian coordinates and the displayed posterior mean and variances. Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model Posterior and marginal-likelihood formulas for the Gaussian sequence prior

[F2]

A square-integrable real variable $Y$ satisfies $P (∣ Y - E Y ∣ \geq u) \leq Var (Y) / u^{2}$ . Chebyshev's inequality for random variables

[F3]

Nonnegative coordinate sums are integrated termwise by Tonelli on the probability space times the counting-measure space. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F4]

Each noise coordinate has law $N (0, 1)$ with density $ϕ (x) = e^{- x^{2} / 2} / \sqrt{2 π}$ and therefore $E Z_{i}^{2} = 1$ . Standard normal and normal laws Location-scale normal laws have their stated mean and variance

Proof

1.1F1F3F4givenalgebra

The posterior coordinate variances satisfy $C^{- 4} i^{2 p} / (i^{q} + n) \leq v_{i} \leq C^{4} i^{2 p} / (i^{q} + n)$ , and the noise weights satisfy $t_{i} = n κ_{i}^{- 2} / (i^{1 + 2 α} κ_{i}^{- 2} + n)^{2} \leq C^{6} n i^{2 p} / (i^{q} + n)^{2}$ . Split at $N$ . For $i \leq N$ , $t_{i} \leq C^{6} i^{2 p} / n$ and $\sum_{i \leq N} i^{2 p} \leq 2 N^{1 + 2 p}$ for $N \geq 2$ . For $i > N$ , $t_{i} \leq C^{6} n i^{- 2 - 4 α - 2 p}$ ; the decreasing-function integral comparison bounds its sum by $2 C^{6} n N^{- 1 - 4 α - 2 p} \leq 2 C^{6} R$ . Thus $\sum_{i} t_{i} \leq 4 C^{6} R$ and $\sup_{i} t_{i} \leq C^{6} R / N$ . By [F4], $E Z_{i}^{2} = 1$ . Two integrations by parts using $ϕ^{'} (x) = - x ϕ (x)$ give $E Z_{i}^{4} = 3$ (all boundary terms vanish), so $Var (Z_{i}^{2}) = 2$ . Independence then bounds the variance of $\sum_{i} t_{i} Z_{i}^{2}$ by $8 C^{12} R^{2} / N$ . Finite sums pass to the nonnegative infinite sum by monotone convergence; the centered sums are Cauchy in $L^{2}$ since $\sum_{i} t_{i}^{2} < \infty$ .

1.2F1F3givenalgebra

Each squared bias coefficient increases with $α$ , so it suffices to consider $b$ . Set $q = q_{b}$ , $N = N_{b}$ . By the bounds on $κ_{i}$ , $∥ B (b) ∥^{2} \leq C^{8} \sum_{i} i^{2 + 4 b + 4 p} θ_{i}^{2} / (i^{q} + n)^{2}$ . For $2 \leq i \leq N$ and $N \geq 4$ , $i^{q + 2 p} / \log i \leq N^{q + 2 p} / \log N$ . Indeed $x^{r} / \log x$ for $r \geq 1$ has its maximum on $[2, N]$ at an endpoint, and its value at $N \geq 4$ exceeds its value at $2$ . The contribution of these indices is consequently at most $C^{8} R h_{n} (b; θ)$ . The index $1$ is zero by hypothesis.

2.1F2step 1.1algebra

All $t_{i} (α)$ decrease in $α$ . Thus the same $Z_{i}$ couple the process and $\sup_{a \leq α \leq b} ∥ W (α) ∥^{2} = \sum_{i} t_{i} (a) Z_{i}^{2}$ . Chebyshev at threshold $R_{a}$ above its mean yields $P {\sup ∥ W ∥^{2} > (4 C^{6} + 1) R_{a}} \leq 8 C^{12} / N_{a}$ , proving (iii).

2.2F1F2step 1.1algebra

For the posterior-radius lower bound, retain only integer indices $N / 2 \leq i \leq N$ . For $N \geq 6$ there are at least $N / 3$ such indices, and each has $v_{i} \geq C^{- 4} 2^{- 2 p - 1} N^{2 p} / n$ . The retained quadratic sum $U_{*}$ therefore has mean at least $d_{0} R$ , where $d_{0} = C^{- 4} 2^{- 2 p - 1} / 3$ . Each retained variance weight is at most $C^{4} N^{2 p} / n = C^{4} R / N$ , so $Var U_{*} \leq 2 (C^{4} R / N) E U_{*}$ . Also $E U_{*} \leq C^{4} R$ , hence $P {U_{*} < d_{0} R / 2} \leq 8 C^{8} / (d_{0}^{2} N)$ . Since $γ \in (0, 1)$ is fixed, the last probability is less than $1 - γ$ for sufficiently large $N$ . The full posterior squared norm dominates $U_{*}$ , so its $1 - γ$ quantile is at least $d_{0} R / 2$ . This proves (i), with $d = d_{0} / 2$ , independently of $A_{n}$ . The full norm is finite a.s. when $α > 0$ because $\sum v_{i} \leq \sum i^{- 1 - 2 α} < \infty$ . At zero use only the defined whole-space branch.

3.1F1step 1.2algebra∎

Put $m = ⌊ N ⌋$ . For large $n$ , $m \geq N_{0}$ and uniformly for $b \leq A_{n}$ , $m^{1 + 2 b} \geq N^{1 + 2 b} / 2$ and $\log m \geq \log N / 2$ , because $(1 + 2 A_{n}) / N_{A_{n}} \to 0$ . On the block $m \leq i \leq ⌊ ρ m ⌋$ , $i^{q} + n \leq (ρ^{q} + 1) n \leq 2 ρ^{q} n$ . Therefore $h_{n} (b; θ) \geq N^{2 b} ρ^{- 2 q} \sum_{i = m}^{⌊ ρ m ⌋} θ_{i}^{2} / 16$ . The polished-tail inequality bounds $\sum_{i > N} θ_{i}^{2}$ by $16 L_{0} ρ^{2 q} R h_{n} (b; θ)$ . Each actual bias coefficient has magnitude at most one, so this bounds the remaining squared bias. Together with step 1.2 and $h_{n} (b) \leq 8 C^{8}$ , this proves (ii).

Depends on

Used by

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

Dependency tree · two levels

37 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

Szabó, van der Vaart and van Zanten, Frequentist coverage of adaptive nonparametric Bayesian credible sets (2015), Section 5 (standard reference, not scraped)