Percy S. Zhai

So Won Jeong, Percy S. Zhai, Veronika Ročková

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.

Boxin Zhao, Percy S. Zhai, Y. Samuel Wang et al.

Undirected graphical models are widely used to model the conditional independence structure of vector-valued data. However, in many modern applications, for example those involving EEG and fMRI data, observations are more appropriately modeled as multivariate random functions rather than vectors. Functional graphical models have been proposed to model the conditional independence structure of such functional data. We propose a neighborhood selection approach to estimate the structure of Gaussian functional graphical models, where we first estimate the neighborhood of each node via a function-on-function regression and subsequently recover the entire graph structure by combining the estimated neighborhoods. Our approach only requires assumptions on the conditional distributions of random functions, and we estimate the conditional independence structure directly. We thus circumvent the need for a well-defined precision operator that may not exist when the functions are infinite dimensional. Additionally, the neighborhood selection approach is computationally efficient and can be easily parallelized. The statistical consistency of the proposed method in the high-dimensional setting is supported by both theory and experimental results. In addition, we study the effect of the choice of the function basis used for dimensionality reduction in an intermediate step. We give a heuristic criterion for choosing a function basis and motivate two practically useful choices, which we justify by both theory and experiments.