stat.MLOct 1, 2026

Posterior sampling by source-space MCMC via prior-based few-step transport maps

Authors: Hoang Phuc Hau Luu, Marcelo Hartmann, Zhongjian Wang

Organizations: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. · Department of Computer Science, Faculty of Science, University of Helsinki, Finland.

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Abstract

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.

Figures & tables

MethodSW2↓\mathrm{SW}_{2}\downarrowBin TV ↓\downarrowCoord. ESS ↑\uparrowCoord. ESS/s ↑\uparrowTime (s) ↓\downarrow
Banana
DPS0.462±0.0710.462\pm 0.0710.311±0.0020.311\pm 0.002——38.4±1.0\mathbf{38.4\pm 1.0}
FM+SPT+pCN0.219±0.0810.219\pm 0.0810.106±0.0080.106\pm 0.008372±16372\pm 160.052±0.0020.052\pm 0.0027087±457087\pm 45
iMF+SPT+pCN0.154±0.0120.154\pm 0.0120.109±0.0060.109\pm 0.006346±4346\pm 42.37±0.122.37\pm 0.12146.1±6.0146.1\pm 6.0
iMF+SPT+hybrid0.142±0.031\mathbf{0.142\pm 0.031}0.085±0.005\mathbf{0.085\pm 0.005}1122±48\mathbf{1122\pm 48}5.33±0.08\mathbf{5.33\pm 0.08}210.4±5.8210.4\pm 5.8
Sine
Table 1 : Synthetic posterior sampling. See Appendix B.2 for more details.
MethodMean CLIP reward ↑\uparrowSampling time (h)Mean sweep time (s)
Golden retriever: “A photograph of a golden retriever running through fallen autumn leaves.”
Best-of-K0.28642.49–
SiT+SPT+pCN0.312210.51370.89
iMF+SPT+pCN0.38092.405.75
iMF+SPT+hybrid0.382410.9526.28
Bald eagle: “A photograph of a bald eagle soaring above snow-capped mountains, wings spread wide.”
Table 2: CLIP reward and measured sampling cost.
Figure 3
Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3 : Marginal joint risk versus squared transport-map error on the two-dimensional banana prior for one- and six-step iMF generation. Points represent training checkpoints, connected in training order; stars mark the final checkpoint. Error bars show one Monte Carlo standard error, conditional on the fitted model and numerical reference, and do not represent variability across training runs.
Figure 4 : Learning the priors by iMF, FM, and diffusion model.
Figure 5 : Posterior sample histograms on the banana prior.
Figure 6 : Posterior sample histograms on the sine prior.
MethodMean CLIP reward ↑\uparrowSampling time (h)Mean sweep time (s)
Golden retriever : “A photograph of a golden retriever running through fallen autumn leaves.”
Best-of-K0.28642.49–
SiT+SPT+pCN0.312210.51370.89
iMF+SPT+pCN0.38092.405.75
iMF+SPT+hybrid0.382410.9526.28
Golden retriever : “A beautiful professional photograph of a golden retriever in a sunlit meadow, sharp focus, natural colors.”
Appendix
Table 3: CLIP reward over retained samples and sampling cost.
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