Conditional Flow Matching for Bayesian Posterior Sampling

We propose a generative multivariate posterior sampling method via flow matching. The method learns a dynamic, block-triangular velocity field in the joint space of data and parameters, which results in a deterministic transport map from a source distribution to the desired posterior. We introduce a time-dependent extension of block-triangular maps for posterior sampling, offering built-in invertibility and scalability. It offers a simple training objective, and does not require the access to likelihood evaluation. The map is automatically invertible without explicit assumptions, a desirable property of the map. The inverse map, named vector rank, is accessible by reversibly integrating the velocity over time. It is advantageous to leverage the dynamic design: proper constraints on the velocity yield a monotone map, which leads to a conditional Brenier map, enabling a fast and simultaneous generation of Bayesian credible sets that agree with Monge-Kantorovich data depth. Our approach is computationally lighter compared to GAN-based and diffusion-based counterparts, and is capable of capturing complex posterior structures. Finally, frequentist theoretical guarantee on the consistency of the recovered posterior distribution, and of the corresponding Bayesian credible sets, is provided.