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Likelihood-free conditional flow matching without an MH correction

Example

Assume the Axiom of Choice. Let a proper prior and forward simulator define a joint prior-predictive law $π$ for an unknown $X_{1}$ and data $Y$ . Suppose the available interface can draw independent pairs $(X_{1}, Y) \sim π$ and reference states $X_{0} \sim η$ , but supplies neither an evaluable likelihood density $L (x_{1}; y)$ nor an auxiliary nonnegative random estimate $\hat{L} (x_{1}; y, U)$ with $E_{U} \hat{L} (x_{1}; y, U) = L (x_{1}; y)$ , where the expectation is over auxiliary randomness $U$ . Conditional flow matching can train from these joint and reference samples and an independent $t \sim Unif [0, 1]$ 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 $λ$ on the standard-Borel state space $R^{d}$ 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 $0 < Z_{y} : = \int_{R^{d}} {\tilde{p}}_{y} (x) λ (d x) < \infty$ , so the posterior is a proper target with finite-valued density $p_{y} = {\tilde{p}}_{y} / Z_{y}$ . Suppose the reference law $η$ 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 $\int q_{y} d λ = 1$ . Its independent proposal kernel is $Q_{y} (x, A) : = \int_{A} q_{y} (z) λ (d z)$ 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 $α_{y} (x, z) = \min ⁣ {1, \frac{{\tilde{p}}_{y} (z) q_{y} (x)}{{\tilde{p}}_{y} (x) q_{y} (z)}} .$ 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.

Example references

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

Facts & Assumptions

Given: A proper prior-predictive simulator for $π$ , independent reference sampling from $η$ , no evaluable likelihood density and no unbiased nonnegative likelihood estimator, and the density-based independent-MH correction specified in the Statement.

[A1]

Assume the Axiom of Choice for the posterior-kernel and density-based Metropolis–Hastings interfaces; its implications give Dependent Choice and Countable Choice for the ODE flow. The Axiom of Choice, AC implies DC implies countable choice, The common-density Metropolis–Hastings ratio

[F1]

Blechschmidt et al. specify that conditional flow matching trains from prior-predictive joint samples and reference draws using a conditional velocity regression objective without evaluating the likelihood. Blechschmidt et al., §3.2 and Remark 3.1

[F2]

The density-based independent-MH correction requires the unnormalized target posterior density in its target/proposal ratio. The common-density Metropolis–Hastings ratio, Owen, Chapter 11 §11.4

[F3]

Under the stated dominated model, the unnormalized posterior density is prior density times likelihood; evidence normalization does not remove the likelihood from an MH ratio. Misfit evidence and the potential posterior law

[F4]

The C $^{1}$ flow clause can supply a proposal density by the Jacobian formula, independently of whether the likelihood density is available. Global Lipschitz ODE flows and the C1 density formula

[F5]

If a distinct exact triangular map satisfies the joint pushforward identity, the conditional transport theorem gives the posterior fiber identity only on a set of full $π_{Y}$ measure; for a fixed query datum the example uses this conclusion only when that datum belongs to this set, unless the fiber identity is separately known there. An exact triangular transport disintegrates to the posterior

[F6]

The Borel state space $R^{d}$ is standard Borel, as required by the density-based Metropolis–Hastings supplier. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel

Verification

Given: [F1]–[F5] and the simulator access limitations in the Statement.

1.1F1givenalgebra

Every training quantity in [F1] is formed from $X_{0}$ , $(X_{1}, Y)$ , independent time, the interpolant and its endpoint difference. These are all available under the specified sampling interface, so no likelihood value is needed to form a CFM training example.

1.2A1F2F3F4givenalgebra

The density-based MH acceptance ratio in [F2] contains the unnormalized posterior target. By [F3] its likelihood contribution is the ratio of likelihoods, which sampling access alone does not supply in general; [F4]'s proposal density does not supply it either. This is an interface limitation rather than an impossibility in every specified model: if the likelihood is independent of the parameter, its ratio is one wherever the forward flow is positive, and the prior and proposal ratios suffice. Thus additional knowledge of an evaluable ratio can permit this correction even without separate likelihood evaluations.

2.1A1F5step 1.1step 1.2given∎

[F5] is a separate route: if the exact joint transport identity is established, the disintegration theorem gives the posterior fiber law on its full-measure set; a fixed query datum must lie in that set unless equality is separately established there. The present example makes no claim that finite training establishes the identity, and supplying a likelihood estimator would change the MH interface.

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Sources

Jan Blechschmidt, Oliver G. Ernst, Moritz Poguntke, and Björn Sprungk, Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems (standard reference, not scraped)
Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 11 §11.4 (standard reference, not scraped)