With the rapid development of rehabilitation robotics, there is a pressing need for efficient and accurate gait prediction methods. However, due to the complexity and variability of individual gait characteristics and external disturbances, accurately predicting gait in real time remains a significant challenge. This paper proposes an innovative Bayesian-inference-based method for real-time gait prediction while a subject walks with a lower-limb exoskeleton. Periodic gait information is represented using von Mises basis functions, and the weight parameters serve as real-time updated state variables. The error-subspace transform Kalman filter (ESTKF) is applied for gait trajectory prediction. A fully connected neural network (FCNN) is used to estimate the walking speeds in real time based on predicted trajectories. Comparative experiments based on an open-source database prove the advantages of ESKTF compared with other Bayesian filters. Walking experiments are conducted to estimate phase and speed in real time, and to predict the joint angle, total joint torque, and lower-limb muscle surface electromyography (sEMG) values. Experimental results validate the method's prediction performance across different speeds and demonstrate its resilience to external interference.