CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Misspecified iid LAN gives a BvM limit when the posterior has the required parameter rate

Statement

Assume the hypotheses of A local KL expansion and square-integrable log-Lipschitz envelope give misspecified iid LAN at $θ^{*}$ , and assume AC. Let a proper prior have a Lebesgue density continuous and strictly positive near $θ^{*}$ , and suppose the posterior has finite positive evidence almost surely and contracts in Euclidean parameter norm outside $n^{- 1 / 2} M_{n}$ for every deterministic $M_{n} \to \infty$ . Here the posterior is based on the iid likelihood $\prod_{i = 1}^{n} p_{θ} (X_{i})$ . Then its law of $\sqrt{n} (θ - θ^{*})$ is $o_{P_{0}} (1) in total variation from N_{d} (V^{- 1} S_{n}, V^{- 1}), S_{n} = n^{- 1 / 2} \sum_{i = 1}^{n} s (X_{i}) .$ The parameter-rate contraction is an explicit premise. Ordinary consistency at a KL projection alone does not discharge it.

Facts & Assumptions

Given: The local smoothness, finite KL-minimizer, score, and contrast hypotheses of the referenced iid LAN theorem; AC; the stated proper prior, evidence, and every-diverging-radius parameter-contraction hypotheses.

[F1]

A local KL expansion and square-integrable log-Lipschitz envelope give misspecified iid LAN: Under its hypotheses, the iid log-likelihood ratio has a compact-uniform $n^{- 1 / 2}$ LAN expansion with central vector $S_{n}$ and positive-definite curvature matrix $V$ .

[F2]

Compact-uniform LAN and rate contraction imply a total-variation BvM limit: Compact-uniform LAN, a prior density continuous and positive at the interior center, proper posterior, and contraction outside $δ_{n} M_{n}$ for every $M_{n} \to \infty$ imply the event-supremum total-variation BvM limit.

Proof

Given: The hypotheses of the two cited theorems and the posterior-rate condition in the Statement.

1.1F1given

By [F1], take $δ_{n} = n^{- 1 / 2}$ and use $S_{n}$ as the central vector $Δ_{n}$ in [F2]. The local LAN theorem supplies the required compact-uniform expansion with curvature $V$ and proves $S_{n} = O_{P_{0}} (1)$ ; [F2] therefore centers the Gaussian at $V^{- 1} S_{n}$ .

2.1F2step 1.1given∎

For each $n$ , the iid sample experiment is dominated by the finite product $μ^{\otimes n}$ , with densities $p_{0, n} (x_{1}, \dots, x_{n}) = \prod_{i} p_{0} (x_{i})$ and $p_{n, θ} (x_{1}, \dots, x_{n}) = \prod_{i} p_{θ} (x_{i})$ . Iterated integration gives the product-law formula on measurable rectangles, so uniqueness of product measures identifies these as densities of $P_{0}^{\otimes n}$ and $P_{θ}^{\otimes n}$ . Thus the LAN expansion in [F1] is the log-likelihood ratio for this iid experiment. The prior, evidence, interior-point, and parameter-tail hypotheses in the Statement are exactly the remaining hypotheses of [F2]. Applying [F2] gives the claimed total-variation limit.

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Sources

B. J. K. Kleijn and A. W. van der Vaart, The Bernstein-Von-Mises theorem under misspecification, Theorem 2.1 and Lemma 2.1 (standard reference, not scraped)
A. W. van der Vaart, Asymptotic Statistics, Chapter 10, §10.2, Theorem 10.1 and proof (standard reference, not scraped)