ProBA: Probabilistic Bundle Adjustment with the Bhattacharyya Coefficient

Classical Bundle Adjustment (BA) is fundamentally limited by its reliance on precise metric initialization and prior camera intrinsics. While modern dense matchers offer high-fidelity correspondences, traditional Structure-from-Motion (SfM) pipelines struggle to leverage them, as rigid track-building heuristics fail in the presence of their inherent noise. We present \textbf{ProBA (Probabilistic Bundle Adjustment)}, a probabilistic re-parameterization of the BA manifold that enables joint optimization of extrinsics, focal lengths, and geometry from a strict cold start. By replacing fragile point tracks with a flexible kinematic pose graph and representing landmarks as 3D Gaussians, our framework explicitly models spatial uncertainty through a unified Negative Log-Likelihood (NLL) objective. This volumetric formulation smooths the non-convex optimization landscape and naturally weights correspondences by their statistical confidence. To maintain global consistency, we optimize over a sparse view graph using an iterative, adaptive edge-weighting mechanism to prune erroneous topological links. Furthermore, we resolve mirror ambiguities inherent to prior-free SfM via a dual-hypothesis regularization strategy. Extensive evaluations show that our approach significantly expands the basin of attraction and achieves superior accuracy over both classical and learning-based baselines, providing a scalable foundation that greatly benefits SfM and SLAM robustness in unstructured environments.