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Motion capture from a monocular video is fundamental and crucial for us humans to naturally experience and interact with each other in Virtual Reality (VR) and Augmented Reality (AR). However, existing methods still struggle with challenging cases involving self-occlusion and complex poses due to the lack of effective motion prior modeling. In this paper, we present a novel variational motion prior (VMP) learning approach for video-based motion capture to resolve the above issue. Instead of directly building the correspondence between the video and motion domain, We propose to learn a generic latent space for capturing the prior distribution of all natural motions, which serve as the basis for subsequent video-based motion capture tasks. To improve the generalization capacity of prior space, we propose a transformer-based variational autoencoder pretrained over marker-based 3D mocap data, with a novel style-mapping block to boost the generation quality. Afterward, a separate video encoder is attached to the pretrained motion generator for end-to-end fine-tuning over task-specific video datasets. Compared to existing motion prior models, our VMP model serves as a motion rectifier that can effectively reduce temporal jittering and failure modes in frame-wise pose estimation, leading to temporally stable and visually realistic motion capture results. Furthermore, our VMP-based framework models motion at sequence level and can directly generate motion clips in the forward pass, achieving real-time motion capture during inference. Extensive experiments over both public datasets and in-the-wild videos have demonstrated the efficacy and generalization capability of our framework.