Posterior sampling by source-space MCMC via prior-based few-step transport maps
Many Bayesian inference problems involve implicit priors known only through samples, such as generative model outputs or historical ensembles. This paper proposes performing posterior sampling in the Gaussian source space of an improved MeanFlow (iMF) transport map, using parallel tempering with pCN or hybrid split-HMC updates. Its central theoretical advance is proving Wasserstein stability bounds for the implemented few-step posterior in terms of the joint population training loss, removing the need for vanishing ODE discretization error.
This paper makes a solid contribution to source-space Bayesian inference by proving that the population training loss of improved MeanFlow controls the Wasserstein error of the implemented few-step posterior, closing a theoretical gap identified in prior flow-matching work. Theorem 3.7 and Corollary 3.10 give an end-to-end bound that vanishes at zero training error without requiring the ODE step size to go to zero, which is a meaningful advance over exact-integration analyses. The hybrid SPT sampler improves mixing on synthetic benchmarks, though its value is less clear on the large image experiment, and the reliance on several strong regularity assumptions with uncontrolled constants tempers the practical guarantee.
The theoretical framing is clean and well-motivated. Under the joint iMF and auxiliary-velocity objective, the authors decompose the marginal risk into population training suboptimality and model-class approximation error via Equation (14), yielding a direct link between the loss minimized in practice and posterior fidelity. Theorem 3.7 is notable because the transport bound vanishes when $\delta=0$ even for a fixed few-step schedule, explicitly addressing the discretization gap left by prior exact-integration results. The synthetic experiments benefit from ground-truth posteriors and show that the proposed iMF-based samplers are more than 30$\times$ faster than a 100-step flow-matching baseline while achieving comparable or lower sliced-Wasserstein and binned-TV errors.
The stability guarantees rest on strong regularity assumptions (Assumptions 3.1–3.5) whose constants—$L_G$, $B_u$, and $C_y(d)$—can depend on the network architecture and scale with dimension in unspecified ways. As the authors acknowledge, "Quantifying how model-class approximation error $\epsilon_{\mathrm{app}}$ decreases with capacity while controlling the regularity constants remains open." In addition, the ImageNet evaluation is limited to two class-prompt pairs in the main text, and the hybrid split-HMC variant offers "no clear improvement in reward or sample quality over iMF+SPT+pCN" despite being a central practical contribution. The coldest-temperature HMC step also introduces sensitivity to gradient noise from the learned few-step map that is not analyzed.
The discussion of related work is candid and precise: the authors correctly identify that Wang et al. (2026b) assume a uniform vector-field bound incompatible with the squared $L^2$ training objective, and that Wang et al. (2026a) bound the exactly integrated flow rather than the discretized map used in practice. They note that "guarantees for the exactly integrated learned flow do not by themselves establish posterior stability for a fixed few-step implementation." The synthetic experiments avoid likelihood-free reference issues by using block-structured targets with available numerical posteriors, making the metric comparisons meaningful. However, the CLIP-guided ImageNet baseline SiT+SPT+pCN was evaluated under a strict matched-time budget because it is "extremely slow," which may understate its potential; and best-of-$K$ is a weak baseline for text-guided generation.
Reproducibility is hampered by the authors' statement that "Our code will be made available soon," meaning no artifacts are released at submission. While synthetic hyperparameters appear in Appendix B.2, the main text omits network depth, learning-rate schedules, and exact training budgets. The ImageNet experiments rely on a pretrained iMF model from an external source, as the authors state: "We use the pretrained iMF for class-conditional ImageNet generation (Geng et al., 2026b)," without specifying how that model was trained or whether it is publicly available. Hardware and wall-clock times are reported, but the lack of open code or data would block an independent full reproduction.
Bayesian inference increasingly uses informative but implicit priors represented only by samples, such as historical ensembles, simulator outputs, and pretrained generative models. The same computational problem appears in the test-time guidance task (generalized Bayes), where an explicit positive weight, e.g., an exponentiated reward, tilts an implicit prior. We develop a framework for source-space generalized Bayesian inference that combines inexpensive few-step prior transports with posterior stability guarantees. Specifically, we represent the prior using a one- or few-step improved MeanFlow (iMF) map and perform posterior sampling in its Gaussian source space. We establish Wasserstein error bounds between the exact and learned posteriors in terms of the joint population iMF and auxiliary-velocity loss, decomposed into training suboptimality and model-class approximation error. In the iMF source space, we adopt parallel tempering with preconditioned Crank-Nicolson updates and introduce a hybrid variant that incorporates split Hamiltonian Monte Carlo to improve sampling efficiency. Synthetic experiments show that the proposed framework can approximate posterior distributions accurately and efficiently, while CLIP-guided ImageNet experiments demonstrate its ability to steer a pretrained iMF image prior toward text-specified preferences.
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