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

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The scaled posterior converges weakly on the negative Sobolev space

Statement

Assume AC and the hypotheses of Weak nonparametric Bernstein–von Mises theorem for product priors. Then the random Borel probability measures ${\tilde{Π}}_{n}$ converge weakly in probability to $N$ on $H (δ)$ , where weak convergence is the bounded-continuous-test definition in Weak convergence of borel probability measures.

Facts & Assumptions

[F1]

Weak nonparametric Bernstein–von Mises theorem for product priors: $β_{B L, H (δ)} ({\tilde{Π}}_{n}, N) \to 0$ in $P_{f_{0}}^{(n)}$ -probability.

[F2]

The bounded-Lipschitz metric characterizes weak convergence on Polish spaces: Under AC, convergence in $β_{B L}$ is equivalent to weak convergence on a Polish space.

[F3]

Weak convergence of borel probability measures: Weak convergence means convergence of integrals for every bounded continuous real test function.

[A1]

The Axiom of Choice: AC is required only for the weak-topology metrization in [F2]; the bounded-Lipschitz conclusion [F1] itself requires only its stated countable-choice premise.

Given: AC and the main theorem's random posterior measures and limit law.

Proof

1.1A1F2

By [F2], the bounded-Lipschitz metric is a metric for the weak topology on the Polish space $H (δ)$ under [A1].

2.1F1F2F3step 1.1∎

The convergence in probability of this metric to zero in [F1] is exactly convergence in probability in the weak topology established in step 1.1; by [F3], its deterministic convergence relation is bounded-continuous-test convergence. This is the asserted weak posterior limit.

Depends on

Used by

Continuous images of the scaled posterior converge weakly Corollary

Dependency tree · two levels

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Sources

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, Definition 1 and Theorem 9, printed pp. 2003–2004 and 2019–2020 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Theorem 7.3.19, printed pp. 595–596 (standard reference, not scraped)