Dimension-uniform conditional flow error for function-valued Bayesian posteriors

Assume the Axiom of Choice. Its implications supply Dependent Choice and Countable Choice for the finite-dimensional ODE flow assumptions.

Let $\mathcal{H}$ be a separable real Hilbert space, let $K\subseteq {\mathbb{R}}^m$ be compact, and let $y\mapsto {\mathrm{\Pi }}^y$ be the pointwise potential posterior of a proper Bayesian inverse model, with positive finite evidence and finite second moment for every $y\in K$. Let $P_n$ be finite-rank orthogonal projections on $\mathcal{H}$ with $P_{n}x\to x$ for every $x\in \mathcal{H}$, let ${\mathcal{H}}_n=P_n\mathcal{H}$, and let ${\mathrm{\Pi }}_n^y$ be proper posteriors defined by declared finite-rank forward/potential approximations on ${\mathcal{H}}_n$, with finite second moments for every $y\in K$. Write $i_n:{\mathcal{H}}_n\hookrightarrow \mathcal{H}$ for inclusion and define the posterior model error $e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t},n}:={\sup }_{y\in K}{W}_{2,\mathcal{H}}(i_{n\mathrm{\#}}{\mathrm{\Pi }}_n^y,{\mathrm{\Pi }}^y),$ where $W_{2,\mathcal{H}}(\mu ,\nu )$ is the infimum, over joint laws with marginals $\mu ,\nu$, of the square root of the expected squared Hilbert distance. This term includes both posterior-potential and finite-rank model error; the known Hellinger potential/discretization estimates would need an explicit moment or tail argument to control it in $W_2$.

Let $\pi$ be the prior-predictive joint law of $(X_1,Y)$ and assume the global prior-predictive moment condition ${\int }_{\mathcal{H}\times {\mathbb{R}}^m}\mathrm{\parallel }{x}_1{\mathrm{\parallel }}_{\mathcal{H}}^2\pi (d{x}_1,dy)<\mathrm{\infty }\mathrm{.}$ Let $\eta$ be an independent reference probability on $\mathcal{H}$ with finite second moment, and use the product training law $\mathrm{Unif}[0,1]\otimes \eta \otimes \pi$. Let $X_t=(1-t)X_0+t{X}_1$ and let $v^{\ast }(t,X_t,Y)=\mathbb{E}[X_1-X_0\mid \sigma (t,X_t,Y)]$ be a chosen continuum conditional regression version. For projected endpoints, define $v_n^{\ast }(t,P_n{X}_t,Y)=\mathbb{E}[P_n(X_1-X_0)\mid \sigma (t,P_n{X}_t,Y)]$ and the finite-rank conditional-target discrepancy $e_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t},n}:={\sup }_{y\in K}{\left({\int }_0^1\mathbb{E}\text{\hspace{0.17em}⁣}[{\parallel P_n{v}^{\ast }(t,X_t,y)-v_n^{\ast }(t,P_n{X}_t,y)\parallel }_{\mathcal{H}}^2\mid Y=y]dt\right)}^{1\mathrm{/}2}\mathrm{.}$ The conditional expectation versions at $Y=y$ must be chosen on all $y\in K$ for these suprema. For a measurable vector field $v:[0,1]\times {\mathcal{H}}_n\times K\to {\mathcal{H}}_n$, define its pointwise conditional population loss by ${\mathcal{L}}_{n,y}(v):={\int }_0^1\mathbb{E}\text{\hspace{0.17em}⁣}[{\parallel v(t,P_n{X}_t,y)-P_n(X_1-X_0)\parallel }_{\mathcal{H}}^2\mid Y=y]dt,$ using the selected conditional-law version. For a finite training sample of size $N$, let ${\bar{v}}_{n,N}$ be an empirical-risk minimizer in the specified vector-field class, when it exists. On realizations where the declared optimizer returns a field (status criterion-met or budget-exhausted), let ${\widehat{v}}_{n,N}$ be that field; the optimizer-failed case has no learned field and is outside this formulation. Define the uniform conditional statistical and optimizer excess-risk terms by $e_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t},n,N}^2:={\sup }_{y\in K}[{\mathcal{L}}_{n,y}({\bar{v}}_{n,N})-{\inf }_{v\in {\mathcal{V}}_n}{\mathcal{L}}_{n,y}(v)],e_{\mathrm{o}\mathrm{p}\mathrm{t},n,N}^2:={\sup }_{y\in K}{[{\mathcal{L}}_{n,y}({\widehat{v}}_{n,N})-{\mathcal{L}}_{n,y}({\bar{v}}_{n,N})]}_{+}$ where ${\mathcal{V}}_n$ is the chosen class and ${\mathcal{L}}_{n,y}$ is its population conditional squared regression loss at datum $y$. Let $e_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r},n,N,h}$ be a certified $W_2$ coupling bound between the numerical endpoint law at solver budget $h$ and the exact learned-flow endpoint law. Finally, let $\mathrm{\Lambda }$ be a proposed bound, uniform in $n$ and $y\in K$, for the time-integrated spatial Lipschitz constants of the learned drifts.

Find checkable assumptions under which the learned finite-rank flow defines a posterior sampler ${\widehat{\mathrm{\Pi }}}_{n,N,h}^y$ and, for each training realization satisfying explicit risk certificates (or with a stated probability over training data), satisfies a dimension-uniform bound of the form ${\sup }_{y\in K}{W}_{2,\mathcal{H}}(i_{n\mathrm{\#}}{\widehat{\mathrm{\Pi }}}_{n,N,h}^y,{\mathrm{\Pi }}^y)\le e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t},n}+e_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r},n,N,h}+C{e}^{\mathrm{\Lambda }}(e_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t},n}+e_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t},n,N}+e_{\mathrm{o}\mathrm{p}\mathrm{t},n,N}),$ with $C$ independent of $n$. The result should derive usable bounds for the posterior model term from the Bayesian potential and discretization hypotheses, establish the uniform-in-$y$ training terms from prior-predictive data, and state the required flow regularity and numerical coupling assumptions. It must not assume that a finite-sample learned field or numerical solver already has the exact joint pushforward property; that property is the conclusion to be certified through the displayed error terms.

The Axiom of Choice implies the choice principles required by the finite-dimensional ODE flow supplier.

The fixed-data posterior result controls Hellinger changes under its common envelope and data-Lipschitz assumptions.

The potential discretization result gives a Hellinger posterior bound under a pointwise potential error envelope.

The posterior ledger is a six-term telescoping identity and does not itself estimate its components.

Finite conditional flow matching trains by regression on prior-predictive samples and returns optimizer and numerical solver statuses.

Blechschmidt et al., Theorem 3.1 and §§3.2–3.4, 5

Zhai et al., §§3.1–3.2

Shikhman, Theorems 23 and 26 and §1.2 scope

Chemseddine et al., §2 and Proposition 1

Status reviewed 2026-10-19 against the inspected full texts. The older broad wording that function-space flow-matching error theory is simply open is stale: Blechschmidt et al. give finite-dimensional inverse-problem CFM theory and function-valued numerical experiments; Zhai, Jeong, and Ročková prove finite-dimensional posterior Wasserstein accuracy in expectation and uniform-in-data consistency under compact-domain, bounded-velocity, smooth-flow and Sobolev assumptions; Shikhman proves generic Hilbert-space functional-flow discretization and end-to-end Wasserstein bounds with sensing, excess-risk, optimizer and solver terms under controlled reconstruction and stability assumptions. Shikhman's stated scope excludes infinite-dimensional likelihoods and arbitrary refinement, while Blechschmidt et al. identify a function-space Wasserstein error analysis as future work. The precise combined posterior-potential, function-space, finite-rank and uniform-conditional result formulated above is not supplied by these inspected sources. This is a bounded source review, not an exhaustive proof that no resolution exists.