Credible sets and frequentist coverage

Fix the inflation $L\ge 1$ in [F2], depending only on the known constants. Let $B_n={\widehat{C}}_n(L)$, including its whole-space zero-selector branch. With $U={\bigcup }_{\beta \in [{\beta }_{-},{\beta }_{+}]}{\mathcal{G}}_{\beta }$, [F2] gives ${ϵ}_n:={\sup }_{\theta \in U}{P}_{\theta }^{(n)}\{\theta \mathrm{\notin }{B}_n\}\to 0$. This is uniform because $U$ is a subset of the single polished-tail class in that theorem. This step claims conservative coverage only.

The random positive-selector posterior is supported on ${\mathrm{\ell }}^2$ on a single Borel domain, not merely on a separate almost-sure domain for each fixed hyperparameter. Indeed intersect the supplied score domain with $T_n=\{x:{\sum }_i{n}^2{\kappa }_i^2{x}_i^2\mathrm{/}{i}^2<\mathrm{\infty }\}\mathrm{.}$ For every square-summable truth the expectation of this nonnegative sum is $n^2{\sum }_i{\kappa }_i^4{\theta }_i^2\mathrm{/}{i}^2+n{\sum }_i{\kappa }_i^2\mathrm{/}{i}^2<\mathrm{\infty }\mathrm{.}$ Thus $T_n$ is Borel and has probability one under every such truth. For every $x\in T_n$ and every $a\ge 0$, the squared posterior mean is bounded by that same sum, since $i^{1+2a}+n{\kappa }_i^2\ge i$. For every $a>0$ the variance trace is finite. Hence all positive-parameter posterior laws at these data are supported on ${\mathrm{\ell }}^2$, including the selected law. Extend the selector by zero and use the whole space outside this common declared domain. This changes none of the sampling probabilities and asserts no posterior on the zero branch.

We verify the diameter rate uniformly in $\beta$, rather than import a pointwise Sobolev assertion. Write $q_a=1+2a+2p$, $N_a=n^{1\mathrm{/}{q}_a}$, and $d=\beta -a\ge 0$. In the score $h_n(a;\theta )$ of [F3], put $e_i=i^{2\beta }{\theta }_i^2$, so ${\sum }_i{e}_i\le R^2$. For $i\le N_a$, the multiplier of $e_i$ is at most $\frac{i^{1-2d}\log i}{N_a\log N_a}\le N_a^{-\min (1,2d)}\mathrm{.}$ If $1-2d\ge 0$, bound both powers and logarithms by their values at $N_a$; otherwise use $i^{1-2d}\le 1$. For $i>N_a$, the multiplier is at most $\frac{N_a^{2q_a}}{N_a\log N_a}{i}^{-1-2a-4p-2\beta }\log i\le N_a^{-2d}\mathrm{.}$ The last function of $i$ is decreasing for $i\ge N_a$ once $n$ is uniformly large, since its negative power has exponent at least $1+4p$. Both ranges therefore give $h_n(a;\theta )\le R^2{N}_a^{-\min (1,2(\beta -a))}\mathrm{.}$ These bounds hold simultaneously for $0\le a\le \beta$, every $\beta$ in the fixed compact interval, and every $\theta \in {\mathrm{\Theta }}_{\beta }(R)$.

Choose a fixed $D>0$ with $R^2\exp \{-2D\mathrm{/}(1+2{\beta }_{+}+2p)\}<t$. For all sufficiently large $n$, put $a_{n,\beta }=\beta -D\mathrm{/}\log n\ge {\beta }_{-}\mathrm{/}2$. For every $0\le a\le a_{n,\beta }$, step 1.3 makes $h_n(a;\theta )<t$: here $\min (1,2(\beta -a))\log N_a\ge 2D\mathrm{/}(1+2{\beta }_{+}+2p)$ when $\log n\ge 2D$. Consequently ${\underset{‾}{\alpha }}_n\ge a_{n,\beta }$, including the empty-threshold convention ${\underset{‾}{\alpha }}_n=A$. The uniform lower bracket [F3] yields ${\sup }_{\beta ,\theta \in {\mathcal{G}}_{\beta }}{P}_{\theta }^{(n)}\{{\widehat{\alpha }}_n<a_{n,\beta }\}⟶0.$ In particular ${\delta }_n:={\sup }_{\theta \in U}{P}_{\theta }^{(n)}\{{\widehat{\alpha }}_n=0\}\to 0$.

Put $H=\{u\in {\mathrm{\ell }}^2:u_1=0\}$, $Z_n=\{{\widehat{\alpha }}_n=0\}$ and $E_n=\{\sqrt{n}{X}_1\le {\mathrm{\Phi }}^{-1}(1-\gamma )\}$. Define $C_n=\{\begin{array}{ll}{\mathrm{\ell }}^2,& Z_n,\\ B_n,& Z_n^c\cap E_n,\\ B_n\setminus H,& Z_n^c\cap E_n^c\mathrm{.}\end{array}$ For $n<3$ one may use ${\mathrm{\ell }}^2$. The positive-selector posterior's first-coordinate variance is $(1+n{\kappa }_1^2{)}^{-1}>0$, so it assigns $H$ mass zero for every data point of the declared domain, even though the hyperparameter is data-selected. Since $L\ge 1$, $B_n$ has posterior mass at least $1-\gamma$; deleting $H$ preserves that mass. On $Z_n$ the specified whole-space convention is retained.

Uniformly for $a\in [{\beta }_{-}\mathrm{/}2,A]$, split the posterior variance trace at $N_a$. Below that cutoff use $v_i(a)\le C^2{i}^{2p}\mathrm{/}n$; above it use $v_i(a)\le i^{-1-2a}$. Summation and an integral comparison give ${\sum }_i{v}_i(a)\le K{n}^{-2a\mathrm{/}{q}_a},$ where $K$ depends only on the fixed constants (the tail factor $1\mathrm{/}(2a)$ is uniformly bounded). Markov's inequality applied to the centered posterior squared norm bounds its $(1-\gamma )$ norm quantile by $\sqrt{K\mathrm{/}\gamma }{n}^{-a\mathrm{/}{q}_a}$. The variances, hence the norm quantile, decrease with $a$ by coupling all centered coordinates with the same standard normals. On the event of step 2.1, the selected radius is therefore at most $\sqrt{K\mathrm{/}\gamma }{n}^{-a_{n,\beta }\mathrm{/}{q}_{a_{n,\beta }}}$. The derivative of $a\mathrm{/}{q}_a$ is $(1+2p)\mathrm{/}{q}_a^2\le 1\mathrm{/}(1+2p)$, so this is at most $K^{\mathrm{\prime }}{n}^{-\beta \mathrm{/}(1+2\beta +2p)}$ with one $K^{\mathrm{\prime }}$ for the full compact $\beta$ interval. Thus outside an event of uniformly vanishing probability, $\mathrm{diam}(B_n)\le 2L{K}^{\mathrm{\prime }}{n}^{-\beta \mathrm{/}(1+2\beta +2p)}$. This includes the uniformly negligible infinite-diameter zero-selector branch.

This is one rule independent of $\theta$ and $\beta$. Its graph is Borel: the coordinate posterior means and selector are Borel, and the squared norm is the limit of its finite coordinate sums. For completeness, on a fixed countable standard-normal product space the map $(a,z)\mapsto {\sum }_i{v}_i(a)z_i^2$ is nonnegative Borel. Integrating its threshold indicators gives a Borel distribution function in $(a,s)$. Its quantile is Borel, by taking the infimum of positive rational radii whose distribution function is at least $1-\gamma$; right continuity ensures this quantile has the required posterior mass. The centered norm has a positive finite quantile: it is finite by its finite second moment and has no atom at zero because its first coordinate is nondegenerate. Thus the radius is positive. Deleting the closed codimension-one hyperplane leaves a dense subset of any positive-radius ball: perturb a point toward the center if necessary and then a sufficiently small amount along the first coordinate. Hence $B_n\setminus H$ has the same diameter $2L{r}_{n,\gamma }({\widehat{\alpha }}_n)$ as $B_n$. Diameter is consequently an extended Borel random variable, equal to this expression off $Z_n$ and infinity on $Z_n$.

Every allowed truth lies in $H$ and has $P_{\theta }^{(n)}(E_n)=1-\gamma$ exactly, since ${\theta }_1=0$. Its coverage equals $P_{\theta }(Z_n)+P_{\theta }(Z_n^c\cap E_n\cap \{\theta \in B_n\})\mathrm{.}$ Subtracting $P_{\theta }(E_n)$ leaves $P_{\theta }(Z_n\cap E_n^c)-P_{\theta }(Z_n^c\cap E_n\cap \{\theta \mathrm{\notin }{B}_n\})\mathrm{.}$ Its absolute value is at most ${\delta }_n+{ϵ}_n\to 0$ uniformly over both suprema in the question. No independence among $E_n$, the selector and $B_n$ is used. Also $C_n\subseteq B_n$ on every branch, so step 3.1 proves the requested diameter statement, in fact with limiting supremum zero for one sufficiently large fixed diameter constant. The nonzero lower norm bound $m$ is retained and needs no additional argument. This answers the unchanged unrestricted existence question affirmatively. It does not identify the constructed sets as balls or attribute this null-deletion construction to the cited literature.