Coverage error under score, forward-solver, and finite-time error

Fix integers $d,n\ge 1$, parameters $p,s,a,R,\sigma >0$, a credibility error $\gamma \in (0,1)$, and a continuous schedule $\beta :[0,T]\to (0,\mathrm{\infty })$ with $T>0$. Let ${\mathrm{\Theta }}_s(R)=\{\theta \in {\mathbb{R}}^d:{\sum }_{j=1}^d{j}^{2s}{\theta }_j^2\le R^2\},A_p=\mathrm{diag}(1^{-p},2^{-p},\dots ,d^{-p})\mathrm{.}$ Observe $Y_i=A_p\theta +{\epsilon }_i$, $1\le i\le n$, with independent ${\epsilon }_i\sim N(0,{\sigma }^2{I}_d)$. Use the proper Gaussian prior ${\mathrm{\Pi }}_0=N(0,\mathrm{diag}(j^{-1-2a}{)}_{j=1}^d)$, and let ${\mathrm{\Pi }}_n(\cdot \mid Y)$ be its exact posterior under the stated forward operator and noise law. Let $m_n(Y)$ be the posterior mean and let $r_{n,\gamma }(Y)$ be the unique radius for which the Euclidean ball $C_{n,\gamma }(Y)=\{u\in {\mathbb{R}}^d:\mathrm{\parallel }u-m_n(Y){\mathrm{\parallel }}_2\le r_{n,\gamma }(Y)\}$ has posterior probability $1-\gamma$. Define the exact frequentist coverage $c_{n,\gamma }(\theta )=P_{n,\theta }\{\theta \in C_{n,\gamma }(Y)\}$ and its baseline error ${\delta }_{n,\gamma }^0={\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mathrm{\mid }{c}_{n,\gamma }(\theta )-(1-\gamma )\mathrm{\mid }$.

For a deterministic numerical matrix ${\widehat{A}}_n\in {\mathbb{R}}^{d\times d}$ with ${\delta }_A\ge 0$ and $\mathrm{\parallel }{\widehat{A}}_n-A_p{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\delta }_A$, let ${\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$ be the Gaussian posterior formed with ${\widehat{A}}_n$ and the same prior and noise variance. Write ${\widehat{p}}_{n,t}^Y$ for the density obtained by applying the forward variance-preserving diffusion $d{X}_t=-\frac{1}{2}\beta (t)X_{t}dt+\sqrt{\beta (t)}d{W}_t$ to $X_0\sim {\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$, and set ${\widehat{s}}_{n,t}^Y=\mathrm{\nabla }\log {\widehat{p}}_{n,t}^Y$. A jointly Borel learned score field $s_{n,t}^Y$ is required to satisfy ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{E}_{Y\sim P_{n,\theta }}[\frac{1}{T}{\int }_0^T{\int }_{{\mathbb{R}}^d}\mathrm{\parallel }{s}_{n,t}^Y(x)-{\widehat{s}}_{n,t}^Y(x){\mathrm{\parallel }}_2^2{\widehat{p}}_{n,t}^Y(x)dxdt]\le {\epsilon }_s^2\mathrm{.}$ For $h=T\mathrm{/}K$, define the reverse-time Euler chain by $Z_0\sim {\widehat{p}}_{n,T}^Y$ and, for $k=0,\dots ,K-1$, $Z_{k+1}=Z_k+h[\frac{1}{2}\beta (T-kh)Z_k+\beta (T-kh)s_{n,T-kh}^Y(Z_k)]+\sqrt{h\beta (T-kh)}{\zeta }_k,$ where the ${\zeta }_k$ are independent $N(0,I_d)$ vectors independent of $Z_0$ and the observations. Draw $M$ independent copies of this chain conditionally on $Y$ and denote their terminal values by $X^{(1)},\dots ,X^{(M)}$. Let $\bar{X}=M^{-1}{\sum }_i{X}^{(i)}$, let $\widehat{r}$ be the empirical $1-\gamma$ quantile of $\{\mathrm{\parallel }{X}^{(i)}-\bar{X}{\mathrm{\parallel }}_2{\}}_{i=1}^M$ (the smallest order statistic whose empirical cumulative mass is at least $1-\gamma$), and put ${\widehat{C}}_{n,K,M}(Y)=\{u:\mathrm{\parallel }u-\bar{X}{\mathrm{\parallel }}_2\le \widehat{r}\}$. Suppose the exact credible-ball boundary satisfies the uniform margin condition ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{P}_{n,\theta }(\mid \mathrm{\parallel }\theta -m_n(Y){\mathrm{\parallel }}_2-r_{n,\gamma }(Y)\mid \le u)\le C_{\mathrm{a}\mathrm{c}}{u}^{\kappa }(0<u\le u_0)$ for fixed $C_{\mathrm{a}\mathrm{c}},u_0>0$ and $\kappa >0$. Determine whether a minimax-sharp nonasymptotic function $\mathrm{\Phi }$ exists uniformly over all such score fields and, if it does, find it as a function of $d,n,p,s,a,R,\sigma ,T,\gamma ,C_{\mathrm{a}\mathrm{c}},u_0,\kappa ,{\delta }_A,{\epsilon }_s,K^{-1}$, and $M^{-1\mathrm{/}2}$ such that ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mid P_{n,\theta ,\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{r}}\{\theta \in {\widehat{C}}_{n,K,M}(Y)\}-(1-\gamma )\mid \le {\delta }_{n,\gamma }^0+\mathrm{\Phi },$ with matching lower bounds that identify when $\mathrm{\Phi }\to 0$. The probability includes observation noise and sampler randomness. The target concerns the additional coverage error introduced by the approximate forward operator, learned score, finite reverse discretization, and finite posterior sample; it does not assume that the exact Bayesian credible ball is already calibrated on every truth class.

Knapik, van der Vaart, and van Zanten, full paper

Szabó, van der Vaart, and van Zanten, full paper

Giné and Nickl, full textbook

Narnhofer et al., full paper

As of 2026-10-19, the full original minimax question is not resolved by this review. A specific obstruction is now proved: with $d=n=R=\sigma =T=1$, $\beta =1$, $\gamma =1\mathrm{/}4$, and zero numerical-operator and integrated score errors, changing the score only at the deterministic Euler grid times can make empirical-ball coverage at truth zero tend to zero along $M=K\to \mathrm{\infty }$. The required uniform boundary-margin condition still holds, and the exact credible-ball baseline error is strictly smaller than $3\mathrm{/}4$. Every valid bound therefore has a positive limiting additional-error term along this admissible scalar path; zero integrated score error and a refined solver grid do not by themselves imply vanishing additional coverage error.

This is a partial answer, not a solution of the full parameter-dependent minimax problem. The original question asks for a sharp function and the regimes in which it vanishes; it does not require it to vanish along every path on which the listed approximation parameters tend to zero. Constant upper bounds exist because every coverage probability lies in $[0,1]$. The proved scalar obstruction does not determine the optimal bound across all original dimensions, sample sizes, credibility levels, margins and error parameters, or characterize every possible vanishing regime. Those parts of the unchanged original formulation remain unanswered here.

The cited Gaussian inverse-sequence papers address exact credible-set coverage and prior/truth dependence. Narnhofer et al.'s conformal error-quantile coverage concerns a different set and does not supply the requested Bayesian credible-ball perturbation bound. No global novelty claim or formulation-specific solution is inferred from the absence of that joint result in these sources.

A scalar zero-integrated-error obstruction to uniform decay along a refined Euler grid. ↗