Conditional flow matching for inverse posteriors — Examples and Counterexamples

Assume the Axiom of Choice for the general density-based Metropolis–Hastings interface. In the stationary-diffusion inverse-problem experiment with $m=256$ observations, noise variance ${\gamma }^2={10}^{-5}$, and $n_{KL}=128$ prior modes, use the baseline conditional-flow-matching MLP with 800 hidden units. The reported flow-matching independent-Metropolis–Hastings acceptance rates for four different true parameters are respectively $0.66\mathrm{\%}$, $11.21\mathrm{\%}$, $5.97\mathrm{\%}$, and $5.47\mathrm{\%}$. In this same experiment, the reported posterior means and 95% bands agree with the DRAM reference to between $0.8\mathrm{\%}$ and $1.8\mathrm{\%}$ in relative $L^2$. The first truth is a prior-predictive tail observation: its normalized radius is $2.53$, the median training radius is $0.85$, and only $0.12\mathrm{\%}$ of the two million training observations have a larger radius. This reported case shows that close posterior means and bands do not by themselves imply high independent-Metropolis–Hastings acceptance or establish full posterior accuracy.

A density-based Metropolis–Hastings correction accepts according to the common-density target/proposal ratio.

Blechschmidt et al., §4.2.1–§4.2.2, Fig. 6 and Table 1

Roberts and Rosenthal, §2.1 and Proposition 2

Assume the Axiom of Choice. Let a proper prior and forward simulator define a joint prior-predictive law $\pi$ for an unknown $X_1$ and data $Y$. Suppose the available interface can draw independent pairs $(X_1,Y)\sim \pi$ and reference states $X_0\sim \eta$, but supplies neither an evaluable likelihood density $L(x_1;y)$ nor an auxiliary nonnegative random estimate $\widehat{L}(x_1;y,U)$ with ${\mathbb{E}}_U\widehat{L}(x_1;y,U)=L(x_1;y)$, where the expectation is over auxiliary randomness $U$. Conditional flow matching can train from these joint and reference samples and an independent $t\sim \mathrm{Unif}[0,1]$ using the linear path $X_t=(1-t)X_0+t{X}_1$ and target velocity $X_1-X_0$, without evaluating $L$. For a fixed query datum $y$, suppose the finite-dimensional prior has a finite-valued Lebesgue density $p_0$ with respect to the sigma-finite Lebesgue measure $\lambda$ on the standard-Borel state space ${\mathbb{R}}^d$ equipped with its Borel sigma-algebra. Assume the model has a finite-valued measurable likelihood density $L(\cdot ;y)$ at this datum, and set ${\tilde{p}}_y(x)=p_0(x)L(x;y)$. Require $0<Z_y:={\int }_{{\mathbb{R}}^d}{\tilde{p}}_y(x)\lambda (dx)<\mathrm{\infty }$, so the posterior is a proper target with finite-valued density $p_y={\tilde{p}}_y\mathrm{/}{Z}_y$. Suppose the reference law $\eta$ has a Lebesgue density and the learned flow satisfies the separate C${}^1$ hypotheses of the ODE lemma; let $q_y$ be a finite-valued measurable version of the resulting proposal density, with $\int q_{y}d\lambda =1$. Its independent proposal kernel is $Q_y(x,A):={\int }_A{q}_y(z)\lambda (dz)$ for every $x$ and Borel $A$. For pairs with positive forward proposal-flow density ${\tilde{p}}_y(x)q_y(z)$, the density-based independent-Metropolis–Hastings acceptance probability is ${\alpha }_y(x,z)=\min \text{\hspace{0.17em}⁣}\{1,\frac{{\tilde{p}}_y(z)q_y(x)}{{\tilde{p}}_y(x)q_y(z)}\}\mathrm{.}$ The stated sampling interface does not in general supply this ratio, so it does not by itself give an evaluable density-based correction. In a specified special model the likelihood ratio may cancel or be otherwise known without separate likelihood evaluations; such a case needs no missing likelihood oracle. This example concerns the general limitation of the specified density-based correction; it does not rule out pseudo-marginal correction if an appropriate likelihood estimator is supplied, nor other correction methods with different inputs.

An exact joint triangular transport gives the posterior conditional law without a likelihood evaluation.

The likelihood-weighted evidence and posterior density are required to form a fixed-data density-based correction.

The density-based Metropolis–Hastings acceptance uses the target and proposal density ratio.

Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel.

The separate C${}^1$ clause supplies a flow proposal density by change of variables when its hypotheses hold.

Blechschmidt et al., §§3.1–3.4 and §4.4

Owen, Chapter 11 §11.4