Industrial big data analysis platform based on the cloud computing is facing the situations like long time-delay of data transmission, large occupation of communication bandwidth, high cost of model training and inference, unsecured data privacy, and etc. However, the arisen edge computing can effectively solve those problems by delegating the power of cloud computing to edge nodes. Aiming at the large-scale industrial processes, the paper proposes a cloud-edge collaborative Variational Bayesian Gaussian Mixture Model (CeC-VBGMM) for distributed process modeling and quality prediction application. In this method, an Alternative Direction Method of Multipliers based Variational Bayesian (ADMM-VB) algorithm is proposed to solve the Gaussian Mixture Model (GMM) for multimode process modeling. The ADMM-VB algorithm provides a natural mechanism to decompose the optimization problem into several subproblems, which can be solved in a distributed and parallel manner across the edge nodes, instead of transferring large-scale data to the server for centralized training. In this way, the industrial datasets can be preserved at local edge nodes to execute the cloud-edge collaborative training, and then a learned global model is formed and saved at the cloud server for real-time model updating at edge nodes. Furthermore, the proposed CeC-VBGMM model is applied to a large-scale industrial process for quality prediction with distributed datasets. Finally, a case study is presented to show the superiorities, comparing with the baselines by centralized training methods.