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

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Continuous images of the scaled posterior converge weakly

Statement

Assume AC and the hypotheses of Weak nonparametric Bernstein–von Mises theorem for product priors. Let $T$ be a metric space and let $G : H (δ) \to T$ be Borel measurable with $N (D_{G}) = 0$ , where $D_{G} : = {x \in H (δ) : G is not continuous at x}$ . For every bounded continuous $h : T \to R$ ,

$\int h d ({\tilde{Π}}_{n} \circ G^{- 1}) ⟶ \int h d (N \circ G^{- 1})$

in $P_{f_{0}}^{(n)}$ -probability. In particular this holds for continuous maps, including continuous linear functionals into $R^{m}$ .

Facts & Assumptions

[F1]

The scaled posterior converges weakly on the negative Sobolev space: ${\tilde{Π}}_{n}$ converges weakly in probability to $N$ on $H (δ)$ .

[F2]

Continuous mapping theorem: If $μ_{n} \Rightarrow μ$ on a metric space and $G$ is measurable with $μ (D_{G}) = 0$ , then $G_{*} μ_{n} \Rightarrow G_{*} μ$ .

[F3]

The bounded-Lipschitz metric characterizes weak convergence on Polish spaces: Under AC, the bounded-Lipschitz metric metrizes weak convergence on Polish $H (δ)$ .

[A1]

The Axiom of Choice: AC is needed only for the weak-topology interpretation imported through [F1] and [F3]; the continuous-mapping step is the choice-free theorem [F2].

Given: AC, a metric target $T$ , the map $G$ , and the posterior and limit measures in the statement.

Proof

1.1A1F2F3given

Fix a bounded continuous $h : T \to R$ and define $Φ (ν) : = \int h d (ν \circ G^{- 1})$ . If $β_{B L, H (δ)} (ν_{m}, N) \to 0$ , [F3] gives $ν_{m} \Rightarrow N$ under [A1]; [F2] and $N (D_{G}) = 0$ imply $ν_{m} \circ G^{- 1} \Rightarrow N \circ G^{- 1}$ , hence $Φ (ν_{m}) \to Φ (N)$ . Since $β_{B L}$ is a metric, this sequential criterion proves continuity of $Φ$ at $N$ .

2.1F1F3step 1.1∎

Given $ε > 0$ , continuity from step 1.1 supplies $η > 0$ such that $β_{B L, H (δ)} (ν, N) < η$ implies $∣ Φ (ν) - Φ (N) ∣ < ε$ . Therefore $P_{f_{0}}^{(n)} {∣ Φ ({\tilde{Π}}_{n}) - Φ (N) ∣ \geq ε} \leq P_{f_{0}}^{(n)} {β_{B L, H (δ)} ({\tilde{Π}}_{n}, N) \geq η} ⟶ 0$ by [F1] and [F3], which give bounded-Lipschitz convergence in probability on $H (δ)$ . This is the stated convergence in probability for $h$ .

Depends on

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Sources

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, continuous-mapping discussion, §4.1, printed pp. 2021–2022 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, continuous-mapping consequence, §7.3.4, printed pp. 600–601 (standard reference, not scraped)