A spiky scale prior defeats an arbitrary degenerate-Gaussian concentration rate

The intervals $I_k$ are disjoint subsets of $[1\mathrm{/}2,3\mathrm{/}2]$, since ${\delta }_{k+1}={\delta }_k^{k+1}$ and ${\delta }_{k+1}+{\delta }_{k+1}^4<{\delta }_k$. The series $S={\sum }_{k\ge k_0}\sqrt{{\delta }_k}$ is positive and finite, bounded by ${\sum }_{k\ge k_0}{e}^{-k\mathrm{/}2}$ using $k!\ge k$ and [F3]. Define ${\mathrm{\Pi }}_{\sigma }=\frac{1}{2}{U}_{[1\mathrm{/}2,3\mathrm{/}2]}+{\sum }_{k\ge k_0}{m}_k{U}_{I_k},m_k=\frac{\sqrt{{\delta }_k}}{2S}\mathrm{.}$ By [F2] this is a probability measure with a measurable Lebesgue density bounded below by $1\mathrm{/}2$ on the required interval. Its spike densities need not be bounded above near $1$.

The zero Gaussian law is tight and centered, with support $\{0\}$, covariance RKHS $\{0\}$ and ${\varphi }_0(ϵ)=0$ for every positive $ϵ$. For all sufficiently large $n$ define ${ϵ}_n=\sqrt{\log \log \log n\mathrm{/}n}$, choosing arbitrary positive initial terms. Then ${ϵ}_n\to 0$, $n{ϵ}_n^2\to \mathrm{\infty }$ and the concentration condition holds. Put $n_k=\lceil k^4{\delta }_k^{-2}\rceil$. Since $\log n_k=2k!+O(\log k)$, $\log \log \log n_k=O(\log k)$ and tends to infinity; hence ${ϵ}_{n_k}\mathrm{/}{\delta }_k=O(\sqrt{\log k}\mathrm{/}{k}^2)\to 0$.

Put $S_n={\sum }_i{Y}_i^2$. By [F1] the likelihood ratio relative to true scale $1$ is $L_n(\sigma )=\exp \{-n\log \sigma +(1-{\sigma }^{-2})S_n\mathrm{/}2\}\mathrm{.}$ On $E_k=\{\mathrm{\mid }{S}_{n_k}-n_k\mathrm{\mid }\le k\sqrt{n_k}\}$, [F4] gives $P_0(E_k^c)\to 0$. Write $q=S_{n_k}\mathrm{/}{n}_k$. Maximizing the log likelihood over all positive $\sigma$ gives its maximum $n_k(q-1-\log q)\mathrm{/}2$ at ${\sigma }^2=q$. Since $\mathrm{\mid }q-1\mathrm{\mid }\le k\mathrm{/}\sqrt{n_k}$, Taylor's formula gives ${\sup }_{\sigma \in [1\mathrm{/}2,3\mathrm{/}2]}{L}_{n_k}(\sigma )\le e^{C{k}^2}$ for a constant independent of $k$.

Uniformly on $I_k$, $\mathrm{\mid }\sigma -1\mathrm{\mid }\le 2{\delta }_k$. The function $d(\sigma )=-\log \sigma +(1-{\sigma }^{-2})\mathrm{/}2$ satisfies $d(1)=d^{\mathrm{\prime }}(1)=0$ and has bounded second derivative near $1$, while $\mathrm{\mid }1-{\sigma }^{-2}\mathrm{\mid }\le C\mathrm{\mid }\sigma -1\mathrm{\mid }$. On $E_k$ this gives $\log L_{n_k}(\sigma )\ge -C{n}_k{\delta }_k^2-Ck\sqrt{n_k}{\delta }_k\ge -C^{\mathrm{\prime }}{k}^4\mathrm{.}$ Thus the posterior denominator is at least $m_k{e}^{-C^{\mathrm{\prime }}{k}^4}$ on $E_k$.

Fix $M>0$. Since ${ϵ}_{n_k}\mathrm{/}{\delta }_k\to 0$, the set $J_k=\{\mathrm{\mid }\sigma -1\mathrm{\mid }\le M{ϵ}_{n_k}\}$ misses $I_k$ and all earlier spikes for large $k$. Its prior mass is at most $M{ϵ}_{n_k}+{\sum }_{j>k}{m}_j$. The later sum is $O(\sqrt{{\delta }_{k+1}})$: successive square-root terms have ratio at most $1\mathrm{/}2$ for $k\ge k_0$, so [F3] bounds the tail by twice its first term. Steps 2.1, 3.1 therefore give ${\mathrm{\Pi }}_{\sigma }(J_k\mid Y^{n_k})\le e^{C{k}^2+C^{\mathrm{\prime }}{k}^4}\frac{M{ϵ}_{n_k}+O(\sqrt{{\delta }_{k+1}})}{m_k}\mathrm{.}$ Both terms tend to zero: ${ϵ}_{n_k}\mathrm{/}\sqrt{{\delta }_k}=O(e^{-k!\mathrm{/}2}\sqrt{\log k}\mathrm{/}{k}^2)$ and $\sqrt{{\delta }_{k+1}\mathrm{/}{\delta }_k}=e^{-kk!\mathrm{/}2}$ dominate the polynomial exponent.

On events with probability tending to one, step 4.1 makes posterior mass within $M{ϵ}_{n_k}$ tend to zero for every fixed $M$. Thus its expected mass outside that radius tends to one, by boundedness and $P_0(E_k^c)\to 0$. The mean function is identically its truth, so this is exactly failure for the combined empirical-mean-plus-scale distance. All original scale-prior and Gaussian concentration assumptions are satisfied, refuting the unrestricted rate claim.