A key consideration in Trajectory Prediction (TP) tools is the confidence that can be placed on the prediction. We purpose a non-deterministic TP neural network using tractable approximate Bayesian variational inference for the model parameters considering weather effects. This work adopts the state-of-art in Bayesian Deep Learning research and builds a neural network model with stochastic convolutional, recurrent, and fully-connect layers. The purposed stochastic variational method outperforms the dropout approximate to Variational Inference and performs reliable uncertainty estimates. It can be easily applied to most neural net architectures and also provides a simple pruning heuristic that can drastically reduce the number of model parameters compares to ensemble methods. The experiment is conducted with the Atlanta Air Route Traffic Control Center (ZTL) flight data and the corridor integrated weather system (CIWS) weather data from Sherlock Data Warehouse (SDW) on June 24th, 2019. The experimental results show better variance reduction than dropout-based methods. The uncertainty estimates are more reliable thanks to the Kullback–Leibler divergence (KL-divergence) term within the optimization objective.