The rapid growth of data volume and velocity is challenging traditional methods of classification, making it impossible to store so much data in memory. Developing online classification methods is becoming increasingly important. Online classification approaches have been subject to several assumptions in recent years, such as an independent and consistent distribution of initial values. In this paper, we present a general online Bayesian classification algorithm that can be adapted to handle streaming data based on the Monte Carlo fusion method. Rather than storing the entire raw data, we only update the posterior with historical data and the current batch of data. In addition, the proposed method has the capability of being applied to a wide range of situations with a high degree of accuracy, even in the case of data imbalances and discrepancies between sub-posterior distributions. The proposed algorithm was validated and evaluated through a comprehensive simulation study.