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A constant GP on a repeated fixed design meets the nondegenerate scale branch

Statement

Take repeated design point $x_{0}$ , prior $W (x) = Z \sim N (0, 1)$ , truth $w_{0} = 0$ , true standard deviation $σ_{0} = 1$ , and an independent prior $σ \sim Unif [1 / 2, 3 / 2]$ . For $ϵ_{n} = 2 \sqrt{\log n / n}$ , the GP concentration and scale-local-mass conditions hold for all sufficiently large n.

Facts & Assumptions

Given: The stated constant Gaussian prior, fixed design, truth and independent uniform scale prior.

[F1]

Standard normal and normal laws: the standard-normal density is bounded below by $φ (1) > 0$ on $[- 1, 1]$ .

[F2]

Gaussian-process posterior contraction in density and regression models: under its fixed-design conditions, with $ρ_{n} = \max {ϵ_{n}, \sqrt{\log n / n}}$ for every allowed prior and with the sharper $ϵ_{n}$ rate for a nonzero Gaussian law, expected posterior mass outside a sufficiently large empirical-L²-plus-scale ball tends to zero.

Verification

1.1F1givenalgebra

The support of $W$ is the line of constant functions, which contains $w_{0}$ . The RKHS approximation cost at zero is zero, and $∥ W ∥_{\infty} = ∣ Z ∣$ . For $ϵ \leq 1$ , [F1] gives $\Pr (∥ W ∥_{\infty} < ϵ) \geq 2 φ (1) ϵ$ , so $ϕ_{0} (ϵ) \leq - \log (2 φ (1) ϵ)$ .

1.2F1givenalgebra

At $ϵ_{n} = 2 \sqrt{\log n / n}$ , $n ϵ_{n}^{2} = 4 \log n$ while $- \log (2 φ (1) ϵ_{n}) = \frac{1}{2} \log n - \frac{1}{2} \log \log n + O (1)$ , so the concentration inequality holds for large n. Also, for $ϵ_{n} < 1 / 2$ , the uniform scale prior assigns mass $2 ϵ_{n}$ to $(1 - ϵ_{n}, 1 + ϵ_{n})$ ; its negative log is $O (\log n) = O (n ϵ_{n}^{2})$ .

2.1F2step 1.1step 1.2∎

The scale prior is independent and has density one, the true scale lies in the interior of its fixed positive interval, and every design point is $x_{0}$ . Steps 1.1–1.2 verify the concentration and scale-local-mass checks, so the sharper fixed-design conclusion follows from [F2]: this Gaussian law is nonzero and the stated rate already satisfies $ρ_{n} = ϵ_{n}$ .

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, §3.3, Theorem 3.3 (standard reference, not scraped)
Subhashis Ghosal and Aad W. van der Vaart, Convergence rates of posterior distributions for noniid observations, §3 (standard reference, not scraped)