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Weak nonparametric Bernstein–von Mises limits — Examples

1 · Prerequisites

2 · Summary

The examples compute the trace threshold for the negative Sobolev noise law and show why white noise cannot be treated as an $L^{2}$ random function. A second example checks every hypothesis for a Gaussian wavelet product prior and an explicit nonzero truth, yielding the stated weak BvM limit.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

White noise has a negative-Sobolev law but no L2-valued law

Example

Take a wavelet basis with $2^{l}$ coefficients at level $l$ and $a_{l} = 2^{l}$ for $l \geq 1$ . The weighted white-noise trace at $δ = 1 / 2$ has level- $l$ contribution

$2^{l} 2^{- l} (\log 2^{l})^{- 1} = \frac{1}{l \log 2},$

so the covariance trace used by this construction diverges. Under countable choice, for every $δ > 1 / 2$ the trace is finite and white noise has a Borel Gaussian law on $H (δ)$ ; under full AC its law is tight. In every infinite-dimensional orthonormal basis, white noise still fails to be an $L^{2}$ -valued random element. This example records the topology and target-law boundary in the weak BvM theorem; it does not assert a separate impossibility theorem for every conceivable $L^{2}$ posterior limit.

Facts & Assumptions

Given: The wavelet-level weights and the standard independent white-noise coordinates.

[F1]

Logarithmic negative Sobolev spaces for white noise: For wavelets the level multiplicity is $2^{l}$ and the weight is $2^{- l} (\log a_{l})^{- 2 δ}$ , with the coarsest finite level treated separately.

[F2]

White noise is a Borel Gaussian law on the negative Sobolev space: Under countable choice, the white-noise series is Borel Gaussian in $H (δ)$ for $δ > 1 / 2$ , while its independent standard-normal coefficients cannot be those of an $L^{2}$ element.

[F3]

The canonical white-noise law is tight under the Axiom of Choice: Under AC, the canonical Borel law on Polish $H (δ)$ is tight.

[A1]

The Axiom of Choice: Full AC is assumed only for the tightness conclusion in [F3]; the trace calculation and independent-coordinate obstruction to an $L^{2}$ realization use no choice axiom.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable choice is assumed for the density-normal calculus interface in [F2]; its Borel Gaussian conclusion is used under this assumption.

Verification

Given: Countable choice, the basis, the weights, and the white-noise coefficient sequence; full AC is additionally assumed for tightness.

1.1F1algebra

At $δ = 1 / 2$ , the $l$ th trace contribution for $l \geq 1$ is $1 / (l \log 2)$ . On each block $2^{m} \leq l < 2^{m + 1}$ its sum is at least $1 / (2 \log 2)$ , so the trace diverges. For $δ > 1 / 2$ the contribution is proportional to $l^{- 2 δ}$ ; on each dyadic block its sum is at most a constant times $2^{m (1 - 2 δ)}$ , a summable geometric sequence. Finite coarsest terms do not affect either conclusion.

1.2A1A2F2F3

For $δ > 1 / 2$ , [F2] under [A2] gives the Borel Gaussian law on $H (δ)$ and the failure of an $L^{2}$ realization; [F3] gives tightness of that law under [A1].

2.1A1A2F2F3step 1.1step 1.2∎

Therefore, under countable choice, the canonical Gaussian target for the weak BvM result is the negative-Sobolev law described in [F2], tight under the additional full-AC branch [F3]; this target-law construction makes no separate claim about posterior convergence in $L^{2}$ or total variation.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

A Gaussian wavelet product prior covered by the weak BvM theorem

Example

Let $φ (u) = (2 π)^{- 1 / 2} e^{- u^{2} / 2}$ be the standard normal density, take any $α > 0$ , and use the product prior scales $σ_{l} = 2^{- l (α + 1 / 2)}$ on an $S$ -regular wavelet basis with $2^{l}$ coefficients at level $l \geq 0$ , as in the weak BvM theorem. Let the truth have coefficients $θ_{0, l k} = σ_{l}$ , so $r_{l k} = 1$ for every $(l, k)$ . With $M = 1$ and $τ = 2$ , all density and coefficient hypotheses of the weak BvM theorem hold.

Facts & Assumptions

Given: The stated $S$ -regular wavelet basis, the standard normal coefficient density, the scale $σ_{l}$ , and the truth coefficients $θ_{0, l k} = σ_{l}$ .

[F1]

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

[F2]

Logarithmic negative Sobolev spaces for white noise: The chosen $S$ -regular wavelet system is a complete orthonormal basis of $L^{2} ([0, 1])$ .

[F3]

Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ : Under countable choice, $L^{2} ([0, 1])$ is complete.

[F4]

Density normal laws and square-summable Gaussian series under countable choice: The standard normal density has mean $0$ and variance $1$ .

[F5]

Weak nonparametric Bernstein–von Mises theorem for product priors: For the stated $S$ -regular wavelet basis, under countable choice and with the supplied white-noise model and posterior kernel, the weak BvM conclusion holds when the truth's normalized coefficients are bounded by $M < τ$ , the prior density is bounded above and bounded below on $(- τ, τ)$ , has finite second moment, and is continuous at each normalized truth coefficient.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable choice supplies the $L^{2}$ completeness in [F3] and the prior construction in [F1]; the full posterior kernel is supplied by the model input of [F5], and no full Axiom of Choice is needed for the bounded-Lipschitz conclusion.

Verification

Given: Countable choice, the stated $S$ -regular wavelet basis, $α > 0$ , $σ_{l} = 2^{- l (α + 1 / 2)}$ , $θ_{0, l k} = σ_{l}$ , $φ (u) = (2 π)^{- 1 / 2} e^{- u^{2} / 2}$ , and the white-noise experiment and supplied full posterior kernel of [F5].

1.1F4givenalgebra

The standard normal density is bounded and continuous, and [F4] gives its second moment $1$ . For $∣ u ∣ < 2$ , $e^{- u^{2} / 2} > e^{- 2}$ , so $φ (u) \geq (2 π)^{- 1 / 2} e^{- 2} > 0$ . The normalized truth coefficients all equal $r_{l k} = 1$ , hence $\sup_{l, k} ∣ r_{l k} ∣ = 1 \leq M = 1 < τ = 2$ .

1.2A1F1F2F3givenalgebra

Let $f_{N} = \sum_{0 \leq l \leq N} \sum_{k \in K_{l}} σ_{l} ψ_{l k}$ . By [F2], for $M > N$ orthogonality gives $∥ f_{M} - f_{N} ∥_{2}^{2} = \sum_{N < l \leq M} 2^{l} σ_{l}^{2} = \sum_{N < l \leq M} 2^{- 2 α l}$ , whose tail tends to zero. Completeness [F3] and [A1] give a limit $f_{0} \in L^{2} ([0, 1])$ ; continuity of each coefficient functional yields $⟨ f_{0}, ψ_{l k} ⟩ = σ_{l}$ . The prior is likewise $L^{2}$ -valued by [F1] under [A1].

2.1A1F2F5givenstep 1.1step 1.2∎

With the stated $S$ -regular basis and supplied posterior kernel, all hypotheses of [F5] now hold, including continuity at each $r_{l k} = 1$ ; therefore for every $δ > 1 / 2$ the scaled posterior converges to canonical white noise in the bounded-Lipschitz metric of $H (δ)$ , in probability under the fixed-truth data law.

Weak nonparametric Bernstein–von Mises limits — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

White noise has a negative-Sobolev law but no L2-valued law

Example

Facts & Assumptions

Verification

A Gaussian wavelet product prior covered by the weak BvM theorem

Example

Facts & Assumptions

Verification

Sources