Visual uncertainties such as occlusions, lack of texture, and noise present significant challenges in obtaining accurate kinematic models for safe robotic manipulation. We introduce a probabilistic real-time approach that leverages the human hand as a prior to mitigate these uncertainties. By tracking the constrained motion of the human hand during manipulation and explicitly modeling uncertainties in visual observations, our method reliably estimates an object's kinematic model online. We validate our approach on a novel dataset featuring challenging objects that are occluded during manipulation and offer limited articulations for perception. The results demonstrate that by incorporating an appropriate prior and explicitly accounting for uncertainties, our method produces accurate estimates, outperforming two recent baselines by 195% and 140%, respectively. Furthermore, we demonstrate that our approach's estimates are precise enough to allow a robot to manipulate even small objects safely.