Bayesian Polarimetric Multi-Source Direction-of-Arrival Estimation for Transient Astronomy with Sparse Radio Interferometric Subarrays

Modern radio interferometers are increasingly challenged by fast transient events, complex radio-frequency interference (RFI), and observing conditions in which near-field and far-field emitters may coexist. Although classical direction-of-arrival (DOA) techniques can achieve high angular resolution, they are often developed for specific array geometries, tend to focus on single-source settings, and usually provide little information about uncertainty. Imaging-based methods, while powerful, are computationally demanding and can introduce delays that are not well suited to real-time transient astronomy. In this work, we propose a Bayesian, polarization-aware framework for multi-source DOA estimation in arbitrary radio interferometric arrays. Starting from baseline-level covariance modeling and polarization-sensitive phase information, we build a probabilistic formulation that jointly infers source direction, polarization state, and, when relevant, source range. Because interferometric phase is inherently wrapped, uncertainty is modeled explicitly using circular statistical distributions, and posterior inference is carried out through a variational Bayesian scheme that remains computationally efficient. Compared with deterministic or purely data-driven approaches, the proposed method offers a more physically grounded and statistically interpretable alternative. It incorporates array geometry, polarization structure, and prior astronomical knowledge directly into the inference process, while also delivering calibrated uncertainty estimates for source localization. Simulations using realistic LOFAR and SKA-Low configurations show robust multi-source separation, stable performance across wide bandwidths, and improved resilience in low signal-to-noise and near-field conditions. Overall, the proposed framework enables imaging-free, uncertainty-aware localization of fast radio bursts, solar radio emission, and terrestrial RFI. It provides a statistically principled and computationally practical route toward real-time transient localization in next-generation radio observatories.