Conditional Flow Matching for Bayesian Posterior Inference
This method offers a computationally efficient alternative to GAN-based and diffusion-based approaches for generating Bayesian credible sets, addressing a domain-specific problem in statistical inference.
The paper tackles Bayesian posterior inference by proposing a generative multivariate posterior sampler using flow matching, which learns a deterministic transport map without requiring likelihood evaluation and provides theoretical guarantees on consistency.
We propose a generative multivariate posterior sampler via flow matching. It offers a simple training objective, and does not require access to likelihood evaluation. 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. 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 whose contours correspond to level sets of 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.