🤖 AI Summary
This paper addresses the challenges in Bayesian posterior inference—namely, reliance on likelihood evaluation, computational inefficiency, and difficulty modeling complex posterior structures. We propose a generative multivariate posterior sampling method based on flow matching. Our approach learns a dynamic block-triangular velocity field in the joint data-parameter space to construct a deterministic Brenier transport map, enabling likelihood-free and efficient posterior sampling. By imposing monotonicity constraints, the map is guaranteed to align with Monge–Kantorovich data-depth level sets, thereby yielding geometrically interpretable Bayesian credible sets endowed with frequentist guarantees—specifically, posterior consistency and credible set convergence. Leveraging conditional flow matching and invertible time integration to solve for the vector field, our method significantly outperforms GANs and diffusion models in computational efficiency while accurately capturing high-dimensional, non-Gaussian, and multimodal posterior geometries.
📝 Abstract
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.