TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Weak nonparametric Bernstein–von Mises theorem for product priors

Statement

Let $(ψ_{l k})$ be the S-regular wavelet basis in Logarithmic negative Sobolev spaces for white noise, with $2^{l}$ coefficients at each level $l \geq 0$ . For $α > 0$ , put $σ_{l} = 2^{- l (α + 1 / 2)}$ and let the prior on $f = \sum_{l, k} σ_{l} ξ_{l k} ψ_{l k}$ have IID coefficients $ξ_{l k}$ with density $φ$ . Assume countable choice for the $L^{2}$ prior construction supplied by The wavelet product prior is L2-valued, and take the full posterior kernel as the supplied model input of The product-prior posterior factors by coordinates. Let $f_{0} \in L^{2} ([0, 1])$ , write $θ_{0, l k} = ⟨ f_{0}, ψ_{l k} ⟩$ and $r_{l k} = θ_{0, l k} / σ_{l}$ , and suppose

$\sup_{l, k} ∣ r_{l k} ∣ \leq M < τ,$

where $φ$ is a probability density satisfying $φ (u) \leq C_{φ}$ for every $u$ , $φ (u) \geq c_{φ} > 0$ for $∣ u ∣ < τ$ , $\int u^{2} φ (u) d u < \infty$ , and $φ$ is continuous at each $r_{l k}$ . The observation model is

$d X^{(n)} (t) = f_{0} (t) d t + n^{- 1 / 2} d W (t), 0 \leq t \leq 1,$

with standard white noise $W$ ; in the stated basis this is the sequence experiment $X_{l k} = θ_{0, l k} + n^{- 1 / 2} Z_{l k}$ with independent standard normal coordinates. Let $Π_{n} (\cdot ∣ X)$ be the product-coordinate posterior of The product-prior posterior factors by coordinates. Let $N$ be the law of the canonical white noise in $H (δ)$ and define

${\tilde{Π}}_{n} : = Π_{n} (\cdot ∣ X) \circ [f \mapsto \sqrt{n} (f - X)]^{- 1}, β_{B L, H (δ)} (μ, ν) : = \sup_{\begin{array}{c} ∥ g ∥_{\infty} \leq 1 \\ {Lip}_{H (δ)} (g) \leq 1 \end{array}} ∣ \int g d μ - \int g d ν ∣ .$

For every $δ > 1 / 2$ ,

$β_{B L, H (δ)} ({\tilde{Π}}_{n}, N) ⟶ 0$

in probability under the true white-noise law $P_{f_{0}}^{(n)}$ . This is a bounded-Lipschitz weak BvM conclusion; it asserts neither total-variation convergence on all Borel subsets nor an $L^{2}$ -valued Gaussian limit.

Facts & Assumptions

[F1]

Logarithmic negative Sobolev spaces for white noise: $H (δ)$ has norm $∥ u ∥_{H (δ)}^{2} = \sum_{l, k} w_{l}^{δ} ∣ u_{l k} ∣^{2}$ , where $w_{l}^{δ} = a_{l}^{- 1} (\log a_{l})^{- 2 δ}$ , and its trace sum is finite for $δ > 1 / 2$ .

[F2]

The wavelet product prior is L2-valued: The product prior series defines an $L^{2}$ -valued random element for $α > 0$ and finite coefficient second moment.

[F3]

White noise is a Borel Gaussian law on the negative Sobolev space: The canonical white-noise coefficient sequence has Borel Gaussian law $N$ on $H (δ)$ for $δ > 1 / 2$ , is not $L^{2}$ -valued, and satisfies $E ∥ W ∥_{H (δ^{'})}^{2} = \sum_{l, k} w_{l}^{δ^{'}} < \infty$ for every $δ^{'} > 1 / 2$ .

[F4]

A negative-Sobolev posterior second-moment bound: For any $δ^{'} > 1 / 2$ , $n E_{f_{0}}^{(n)} \int ∥ f - f_{0} ∥_{H (δ^{'})}^{2} Π_{n} (d f ∣ X)$ is bounded uniformly in $n$ .

[F5]

Finite-coordinate posterior total-variation limit: For every fixed finite coordinate projection, the posterior and standard Gaussian densities of $\sqrt{n} (θ_{J} - X_{J})$ converge in $L^{1}$ in probability.

[F6]

The product-prior posterior factors by coordinates: A supplied posterior kernel has product-coordinate marginals with densities proportional to $e^{- n (θ_{j} - X_{j})^{2} / 2} σ_{j}^{- 1} φ (θ_{j} / σ_{j})$ .

[F7]

Cauchy-Schwarz for random variables: On any probability space, $E ∣ U V ∣ \leq (E U^{2})^{1 / 2} (E V^{2})^{1 / 2}$ for square-integrable real random variables.

[F8]

Markov's inequality for random variables: For nonnegative $Y$ and $a > 0$ , $P (Y \geq a) \leq E [Y] / a$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable choice supplies the $L^{2}$ prior construction and the density normal-law interface used by [F3]. The full posterior kernel is supplied as model input by [F6]; the posterior moment estimate, finite-coordinate convergence, and projection/tail inequalities introduce no further choice axiom.

Given: Countable choice, the basis, prior, truth, density conditions, white-noise experiment, and the supplied posterior kernel of [F6]. The stated bounded-Lipschitz convergence is proved below without using full AC.

Proof

1.1F1givenalgebra

Fix $δ > 1 / 2$ and choose $δ^{'}$ with $1 / 2 < δ^{'} < δ$ . Let $P_{J}$ retain all wavelet coordinates with level $l \leq J$ . Since $w_{l}^{δ} / w_{l}^{δ^{'}} = (\log a_{l})^{- 2 (δ - δ^{'})}$ and $a_{l} \to \infty$ , for $q_{J} : = (\log a_{J + 1})^{- (δ - δ^{'})}$ one has $∥ (I - P_{J}) u ∥_{H (δ)} \leq q_{J} ∥ u ∥_{H (δ^{'})} and q_{J} \to 0.$ For any probability law $Q$ on $H (δ)$ , coupling $U$ with $P_{J} U$ and using the defining Lipschitz bound gives $β_{B L, H (δ)} (Q, Q \circ P_{J}^{- 1}) \leq \int ∥ (I - P_{J}) u ∥_{H (δ)} Q (d u)$ .

1.2F5algebra

For a fixed $J$ , the middle term in $β_{B L, H (δ)} ({\tilde{Π}}_{n}, N) \leq β_{B L, H (δ)} ({\tilde{Π}}_{n}, {\tilde{Π}}_{n} \circ P_{J}^{- 1}) + β_{B L, H (δ)} ({\tilde{Π}}_{n} \circ P_{J}^{- 1}, N \circ P_{J}^{- 1}) + β_{B L, H (δ)} (N \circ P_{J}^{- 1}, N)$ tends to zero in probability: [F5] gives $L^{1}$ convergence of the projected posterior and Gaussian coordinate densities. Every bounded-Lipschitz test function restricted to the projected subspace is bounded by $1$ , so its integral difference is at most that $L^{1}$ distance.

1.3F1F3F7algebra

The last term is bounded by $E ∥ W - P_{J} W ∥_{H (δ)}$ , which is at most $(E ∥ W - P_{J} W ∥_{H (δ)}^{2})^{1 / 2} = (\sum_{l > J, k} w_{l}^{δ})^{1 / 2}$ by [F3] and [F7]. This tends to zero as $J \to \infty$ by [F1].

2.1A1F2F3F4F6F7step 1.1algebra

Under the joint true-data/posterior law, using the supplied posterior model [F6] and the $L^{2}$ prior construction [F2], $U = \sqrt{n} (f - X) = \sqrt{n} (f - f_{0}) - W$ , where $W = \sqrt{n} (X - f_{0})$ has law $N$ . By [F4], [F3] and $(a - b)^{2} \leq 2 a^{2} + 2 b^{2}$ , $E_{f_{0}}^{(n)} \int ∥ U ∥_{H (δ^{'})}^{2} Π_{n} (d f ∣ X) \leq 2 n E_{f_{0}}^{(n)} \int ∥ f - f_{0} ∥_{H (δ^{'})}^{2} Π_{n} (d f ∣ X) + 2 E ∥ W ∥_{H (δ^{'})}^{2} \leq C,$ with $C$ independent of $n$ . Applying [F7] under the posterior probability and then under the true-data probability, together with step 1.1, gives $E_{f_{0}}^{(n)} β_{B L, H (δ)} ({\tilde{Π}}_{n}, {\tilde{Π}}_{n} \circ P_{J}^{- 1}) \leq q_{J} \sqrt{C} .$ Thus the first projection error can be made uniformly small by taking $J$ large.

3.1F8step 2.1step 1.2step 1.3algebra∎

Fix $0 < ε < 1$ and choose $J$ so the expected first and deterministic third terms are each less than $ε^{2}$ . By [F8], the probability that the first term exceeds $ε$ is at most $ε$ ; the deterministic third term is less than $ε$ . For this fixed $J$ , step 1.2 makes the probability that the middle term exceeds $ε$ tend to zero. The triangle bound then gives a limiting upper bound $ε$ for the probability that the full distance exceeds $3 ε$ . Since $ε$ is arbitrary, $β_{B L, H (δ)} ({\tilde{Π}}_{n}, N) \to 0$ in $P_{f_{0}}^{(n)}$ -probability.

Depends on

Used by

Dependency tree · two levels

31 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

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, Theorems 5, 7–9 and proofs, printed pp. 2013–2021 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Theorems 7.3.12, 7.3.18–7.3.19 and proofs, printed pp. 584–596 (standard reference, not scraped)
Ray, Adaptive Bernstein–von Mises theorems in Gaussian white noise, §§2.1–2.2 and §7.1, printed pp. 5–8 and 23–24 (standard reference, not scraped)