Point cloud data have become increasingly vital due to their ability to capture detailed, multi-dimensional representations of physical objects and their surrounding environments. Point clouds are nowadays central to applications across industry and academia that range from robotics, navigation systems, and 3D printing to architecture, manufacturing, and agriculture. Despite its importance, the analysis of point cloud data faces several limitations, including high computational cost due to large data volume, contamination with localization noise and inaccuracies caused by sensor limitations, and missing points due to environmental factors or sensor positioning. Moreover, unavailable uncertainty quantification in reconstructed structures remains a critical gap in applications requiring reliable estimation. We present a fully Bayesian framework for point cloud data analysis and curve reconstruction. Our framework models a cloud’s points as noisy perturbations of latent positions constrained to lie on closed polylines, jointly inferring polyline vertices and connectivity, latent coordinates, and noise characteristics. Posterior inference is performed via a specialized Markov chain Monte Carlo sampler tailored to point cloud data processing. Experiments on synthetic data demonstrate accurate curve reconstruction while providing uncertainty quantification.