Conditional flow matching for inverse posteriors

Assume the Axiom of Choice. Let $d,m,n,B\ge 1$, let $\eta$ be a reference probability on ${\mathbb{R}}^d$, and let $\pi$ be the proper joint law of an unknown $X_1\in {\mathbb{R}}^d$ and data $Y\in {\mathbb{R}}^m$ in a specified Bayesian inverse model. Write ${\pi }_Y$ for the data marginal and ${\pi }_y$ for a posterior-kernel version of the conditional law of $X_1$ given $Y=y$. Assume ${\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^m}\mathrm{\mid }{x}_1{\mathrm{\mid }}^2\pi (d{x}_1,dy)+{\int }_{{\mathbb{R}}^d}\mathrm{\mid }{x}_0{\mathrm{\mid }}^2\eta (d{x}_0)<\mathrm{\infty }\mathrm{.}$ Choose a jointly measurable parameterized vector-field family $v_{\theta }:[0,1]\times {\mathbb{R}}^d\times {\mathbb{R}}^m\to {\mathbb{R}}^d$, with parameter set $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$ for some $p\ge 1$, whose candidates have finite population squared loss under the training law below.

Declared inputs. Supply independent sampling routines for $\mathrm{Unif}[0,1]$, $\eta$, and $\pi$; here $\mathrm{Unif}[0,1]$ is Lebesgue probability on the unit interval. Supply an initialiser $\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}$ returning a parameter in $\mathrm{\Theta }$ or failure; an optimizer update callable $\mathrm{U}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}(k,\theta ,{\widehat{\mathcal{L}}}_n,{\mathcal{D}}_n)$ returning a parameter in $\mathrm{\Theta }$ or failure, with each call counting as one optimizer step; a named stopping statistic callable $S(k,\theta ,{\widehat{\mathcal{L}}}_n,{\mathcal{D}}_n)$ returning a finite real value when evaluation succeeds; a finite step budget $M\ge 1$ and tolerance ${\epsilon }_{\mathrm{o}\mathrm{p}\mathrm{t}}>0$; a flow-check callable; and a numerical ODE solver with a declared finite step or tolerance budget $h$. Every callable must return its specified value or a failure status in finite time. Any randomization used by these callables is part of the declared input.

Training and optimizer instructions. Draw $n$ independent triples $(t_i,X_{0i},(X_{1i},Y_i))\sim \mathrm{Unif}[0,1]\otimes \eta \otimes \pi ,$ and form $X_{t_i}=(1-t_i)X_{0i}+t_i{X}_{1i}$ and $V_i=X_{1i}-X_{0i}$. Let ${\mathcal{D}}_n=((t_i,X_{t_i},Y_i,V_i){)}_{i=1}^n$ and evaluate the empirical squared loss ${\widehat{\mathcal{L}}}_n(\theta )=\frac{1}{n}{\sum }_{i=1}^n\mid v_{\theta }(t_i,X_{t_i},Y_i)-V_i{\mid }^2\mathrm{.}$ If sampling fails, return sampling-failed. Otherwise call $\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}$; if it fails or returns a parameter outside $\mathrm{\Theta }$, return optimizer-failed with no fitted field. Set $k=0$ and repeat the following steps in order: evaluate ${\widehat{\mathcal{L}}}_n({\theta }_k)$ and $S(k,{\theta }_k,{\widehat{\mathcal{L}}}_n,{\mathcal{D}}_n)$; if either evaluation fails or either value is nonfinite, return optimizer-failed with no fitted field; if $S\le {\epsilon }_{\mathrm{o}\mathrm{p}\mathrm{t}}$, stop with status criterion-met; if $k=M$, stop with status budget-exhausted; otherwise call $\mathrm{U}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}(k,{\theta }_k,{\widehat{\mathcal{L}}}_n,{\mathcal{D}}_n)$, returning optimizer-failed with no fitted field if the call fails, returns a parameter outside $\mathrm{\Theta }$, or yields a nonfinite value, and then set ${\theta }_{k+1}$ to its output and increase $k$ by one. For either successful stopping status, return $\widehat{v}=v_{{\theta }_k}$, its achieved training loss, the last value of $S$, $k$, $M$, ${\epsilon }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ and the optimizer status. A finite-data stopping status does not certify $\widehat{v}=v^{\ast }$.

The population least-squares target is the conditional-expectation class $v^{\ast }(t,x,y)=\mathbb{E}[X_1-X_0\mid \sigma (t,X_t,Y)]$ coordinatewise, evaluated at $(t,x,y)$ through a chosen measurable representative. A conditional expectation is only an almost-sure equivalence class, so this formula does not specify an off-support pointwise extension.

Flow and sampling instructions. For a query datum $y$, call the flow checker on $\widehat{v}$ and the data-parameter domain containing $y$. It returns pass only when it certifies all hypotheses of the ODE flow lemma, including joint continuity and the integrable global spatial-Lipschitz and origin-growth bounds on compact data-parameter sets containing $y$. If it does not return pass, return field-regularity-failed with the optimizer record and no terminal samples. Otherwise, for $j=1,\dots ,B$, draw independent $Z_0^j\sim \eta$ and call the declared solver to integrate on $0\le t\le 1$ ${\dot{Z}}_t^j=\widehat{v}(t,Z_t^j,y),Z_0^j,$ with budget $h$. Record the returned finite endpoint as the terminal approximation ${\widehat{Z}}_1^j$ only when the solver reports success; treat a success response with a nonfinite endpoint as a failed integration. Also record each status and any certified solver error estimate, marking an unavailable estimate unquantified. Continue through all $B$ calls after a solver failure. Return the successful terminal approximations and all per-call records, together with the field and optimizer records; the overall status is completed if every integration succeeds and solver-failed otherwise. A sampling-call failure returns sampling-failed and the records collected so far.

If the exact triangular flow of $\widehat{v}$ were known to satisfy $T_{\mathrm{\#}}(\eta \otimes {\pi }_Y)=\pi$, the exact transport theorem would imply $T_y{}_{\mathrm{\#}}\eta ={\pi }_y$ for ${\pi }_Y$-almost every $y$. Finite regression loss, optimizer termination, and solver success alone do not establish this joint pushforward identity or exact posterior sampling. Record posterior model, discretization, kernel, burn-in, Monte Carlo and solver components under the declared posterior-computation error budget when applicable. The budget does not supply a learned-flow regression or optimizer-error estimate; report those separately as measured or certified quantities, and mark unavailable quantities unquantified rather than zero.

A proper prior and likelihood model induce a joint prior-predictive law whose data-indexed conditional kernel is the posterior version.

Conditional expectation is the measurable integrable representative characterized by its integrals on the conditioning sigma-algebra.

Under AC, conditional expectation is the unique almost-sure least-squares minimizer in the conditioning $L^2$ space.

On standard Borel conditioning coordinates, a conditional expectation is represented by a measurable function of those coordinates.

The integrable global Lipschitz and growth hypotheses give a unique global ODE flow and quantitative bi-Lipschitz bounds.

An exact joint triangular pushforward gives the conditional posterior pushforward almost everywhere.

The posterior error ledger has six adjacent model, solver, discretization, invariant-kernel, burn-in and Monte Carlo terms, but no learned-flow training term.

Blechschmidt et al., §§3.1–3.2 and Appendix B

Tong et al., Theorem 3.2 and §3.2

Zhai et al., §§3.1–3.2