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A degenerate GP concentration function does not bound unknown-scale prior mass

Statement

Let $W \equiv 0$ , $w_{0} = 0$ , and let its covariance RKHS be $H = {0}$ . Define $ϕ_{0} (ϵ) = \inf_{h \in H, ∣ h - w_{0} ∣ < ϵ} {∥ h ∥_{H}^{2}} - \log \Pr (∣ W ∣ < ϵ)$ . For $n$ large, set $ϵ_{n} = \sqrt{\log \log n / n}$ and take an independent $σ \sim Unif [1 / 2, 3 / 2]$ with true value $σ_{0} = 1$ . Then $ϕ_{0} (ϵ_{n}) = 0$ , $ϵ_{n} \to 0$ , and $n ϵ_{n}^{2} \to \infty$ , but $- \log \Pr (∣ σ - σ_{0} ∣ < ϵ_{n}) ≠ O (n ϵ_{n}^{2}) .$ Thus the Gaussian concentration condition alone does not verify the joint local prior-mass bound used by the denominator argument for the unknown-scale branch. This calculation is a proof-gap example, not a posterior counterexample.

Facts & Assumptions

Given: The point-mass Gaussian prior, the displayed rate sequence, and the independent uniform scale prior.

[F1]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log x = \int_{1}^{x} d t / t$ and, for $x \geq 1$ , $\log x \leq 2 \sqrt{x}$ .

Verification

1.1givenalgebra

The RKHS of the point-mass prior is ${0}$ ; its approximation cost at $w_{0} = 0$ is zero. Also $\Pr (∣ W ∣ < ϵ) = 1$ for every $ϵ > 0$ , so the explicitly defined concentration function satisfies $ϕ_{0} (ϵ) = 0$ .

1.2givenalgebra

For $n$ sufficiently large, $ϵ_{n} < 1 / 2$ , hence the uniform prior assigns mass $2 ϵ_{n}$ to $(1 - ϵ_{n}, 1 + ϵ_{n})$ . Therefore $- \log (2 ϵ_{n}) = \frac{1}{2} \log n - \frac{1}{2} \log \log \log n - \log 2,$ while $n ϵ_{n}^{2} = \log \log n$ . Along $n_{m} = ⌈ e^{e^{m}} ⌉$ , the first quantity grows on the order of $e^{m}$ and the second on the order of $m$ , so their ratio is unbounded.

2.1F1step 1.2given∎

Since $\log \log n \to \infty$ and $\log \log n / n \to 0$ , the rate has $n ϵ_{n}^{2} \to \infty$ and $ϵ_{n} \to 0$ while the scale interval mass has negative logarithm not $O (n ϵ_{n}^{2})$ by step 1.2. Thus the Gaussian concentration condition does not establish this particular denominator-mass check; it does not establish failure of posterior contraction. Any separate posterior-contraction argument requires a specified sampling likelihood and its own hypotheses; a local bound on the scale-prior density alone is insufficient.

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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 (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)