In contrast to the assumptions of most existing Electromigration (EM) analysis tools, the evolution of EM-induced stress is inherently non-deterministic, influenced by factors such as input current fluctuations and manufacturing non-idealities. Traditional approaches for estimating stress variations typically involve computationally expensive and inefficient Monte Carlo simulations with industrial solvers, which quantify variations using mean and variance metrics. In this work, we introduce a novel machine learning-based framework, termed BPINNEM- Post, for efficient stochastic analysis of EM-induced postvoiding aging processes. This new approach integrates closedform analytical solutions with a Bayesian Physics-Informed Neural Network (BPINN) framework to accelerate the analysis for the first time. The closed-form solutions enforce physical laws at the individual wire segment level, while the BPINN ensures that physics constraints at inter-segment junctions are satisfied and stochastic behaviors are accurately modeled. By reducing the number of variables in the loss functions through the use of analytical solutions, our method significantly improves training efficiency without accuracy loss and naturally incorporates variational effects. Additionally, the analytical solutions effectively address the challenge of incorporating initial stress distributions in interconnect structures during post-void stress calculations. Numerical results demonstrate that BPINN-EM-Post achieves over 240x speedup compared to Monte Carlo simulations using the FEM-based COMSOL solver and more than 65x speedup compared to Monte Carlo simulations using the FDM-based EMSpice method.