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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.

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Sources

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, Condition 1 and Theorem 9, printed pp. 2013–2014 and 2019–2020 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Condition 7.3.11 and Theorem 7.3.19, printed pp. 584 and 595–596 (standard reference, not scraped)