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The integral functional under a product-prior white-noise BvM

Example

Take a sufficiently regular Cohen–Daubechies–Vial wavelet basis as in Weak nonparametric Bernstein–von Mises theorem for product priors, with constants contained in a finite coarsest scaling space and all finer wavelets orthogonal to constants. Let $f_{0} = 0$ , use independent standard-normal product coefficients, and choose any prior scale exponent $η > 0$ with $σ_{l} = 2^{- l (η + 1 / 2)}$ . The functional $L_{1} (f) = \int_{0}^{1} f (x) d x$ has asymptotically standard-normal posterior fluctuations centered at $L_{1} (X^{(n)})$ , and its equal-tailed posterior interval has asymptotic frequentist coverage $1 - α$ .

Facts & Assumptions

Given: The white-noise model, $f_{0} = 0$ , the selected regular CDV basis, independent standard-normal prior coefficients scaled by $2^{- l (η + 1 / 2)}$ , and the supplied posterior kernel in Weak nonparametric Bernstein–von Mises theorem for product priors.

[F1]

Smooth linear functionals on the white-noise topology: $L_{1} (f) = ⟨ f, 1 ⟩_{L^{2}} = \int_{0}^{1} f$ .

[F2]

Linear-functional BvM limit and credible-interval calibration: any nonzero $g \in H_{2}^{s}$ , $s > 1 / 2$ , yields a scalar $N (0, ∥ g ∥_{2}^{2})$ posterior BvM limit and asymptotic $1 - α$ coverage for the equal-tailed interval under the product-prior theorem's hypotheses.

[F3]

Weak nonparametric Bernstein–von Mises theorem for product priors: the prior uses scales $2^{- l (α + 1 / 2)}$ and requires a bounded density positive near the normalized true coefficients, finite coefficient second moment, the coefficient bound, continuity at those coefficients, and countable choice for the $L^{2}$ prior construction.

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice supplies a choice function for a countable family of nonempty sets; here it is the axiom explicitly assumed for the product-prior construction.

[A1]

For the selected CDV basis, constants lie in the finite coarse scaling space and every wavelet above its finite base level is orthogonal to constants.

Verification

Given: The hypotheses in the Example and Facts & Assumptions, with countable choice for the $L^{2}$ prior construction.

1.1F3givenalgebra

The standard-normal coefficient density is bounded, continuous, strictly positive at every real value, and has finite second moment. Since $f_{0} = 0$ , every normalized true coefficient is zero; choose any $0 < M < τ$ with $τ > 0$ . The coefficient bound and continuity at each normalized true coefficient required by [F3] hold, and the scale sequence has the theorem's required form with $α = η > 0$ .

1.2A1F1givenalgebra

By [A1], all wavelet coefficients of $g = 1$ above a fixed finite level vanish; the remaining levels have finitely many coefficients. Therefore $\sum_{l, k} a_{l}^{2 s} ∣ g_{l k} ∣^{2} < \infty$ for every finite $s$ , in particular for some $s > 1 / 2$ . Also $∥ g ∥_{2} = 1$ and its pairing is $L_{1} (f) = \int_{0}^{1} f$ .

2.1F2F3F4givenalgebra∎

Under $f_{0} = 0$ and countable choice [F4], $\sqrt{n} L_{1} (X^{(n)}) = L_{1} (W)$ is exactly $N (0, 1)$ . Applying [F2] with $g = 1$ gives the scalar posterior BvM limit $N (0, 1)$ and limiting coverage $1 - α$ .

Depends on

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Sources

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, Condition 1 and Theorem 3 (standard reference, not scraped)
Castillo and Nickl, On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures, §2.2 (standard reference, not scraped)