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How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A constant Gaussian log-density prior gives the uniform density exactly

Statement

On $[0, 1]$ with Lebesgue measure, take $W (x) = Z$ for all $x$ , where $Z \sim N (0, 1)$ , and let the truth be $w_{0} = 0$ . The normalized log-Gaussian density prior is exactly the point mass at the uniform density. The concentration condition holds at $ϵ_{n} = \sqrt{\log n / n}$ for all sufficiently large n.

Facts & Assumptions

Given: The constant process $W (x) = Z$ , standard normal $Z$ , and uniform truth $p_{0} = 1$ .

[F1]

Standard normal and normal laws: the standard normal has density $φ (z) = e^{- z^{2} / 2} / \sqrt{2 π}$ , which is bounded below on $[- 1, 1]$ by $φ (1) > 0$ .

[F2]

Gaussian-process posterior contraction in density and regression models: in the log-Gaussian branch, a verified concentration condition and the model hypotheses imply expected posterior contraction in Hellinger distance.

Verification

1.1givenalgebra

Every prior path is constant, so $Z_{W} = \int_{0}^{1} e^{Z} d x = e^{Z}$ and $p_{W} (x) = 1 = p_{0} (x)$ . Thus the posterior distribution of the density itself is the point mass at $p_{0}$ for every data vector; its posterior mass outside every positive Hellinger ball is exactly zero.

1.2F1givenalgebra

The RKHS approximation term at $w_{0} = 0$ is zero because the zero vector belongs to every Hilbert space. For $0 < ϵ \leq 1$ , [F1] gives $\Pr (∥ W ∥_{\infty} < ϵ) = \Pr (∣ Z ∣ < ϵ) \geq 2 φ (1) ϵ$ , hence $ϕ_{0} (ϵ) \leq - \log (2 φ (1) ϵ)$ . For $ϵ_{n} = \sqrt{\log n / n}$ , $n ϵ_{n}^{2} = \log n$ and $- \log ϵ_{n} \leq \frac{1}{2} \log n$ for large n, so $ϕ_{0} (ϵ_{n}) \leq n ϵ_{n}^{2}$ eventually.

2.1F2step 1.1step 1.2∎

The density family is jointly measurable and all normalizers are positive and finite. Step 1.2 verifies the concentration premise; applying the log-Gaussian branch of [F2] gives the stated rate, while step 1.1 gives the stronger exact posterior conclusion.

Depends on

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Sources

A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors, §3.1 (standard reference, not scraped)
Bas J. K. Kleijn, Aad W. van der Vaart and Harry van Zanten, Lectures on Nonparametric Bayesian Statistics, §12.2 (standard reference, not scraped)