We introduce a Bayesian generative formulation of similarity-weighted aggregation (SimAgg) to stochastically combine the weights from privacy-compliant federated collaborators for brain lesion segmentation. Bayesian SimAgg can adapt effectively to the variability in data across federated collaborators and leverages probabilistic modeling to account for uncertainty in the model parameters. It ensures a robust and flexible aggregation of the federated models by taking into account the central tendency and the inherent variability of the collaborators. Our Bayesian formulation employs multi-armed bandit in a novel setup to optimally select participating collaborators suitable for a dynamic federated learning (FL) environment. In simulation experiments using multi-parametric MRI data from 1,251 glioblastoma cases from the Federated Tumor Segmentation (FeTS) 2022 challenge, Bayesian SimAgg with UCB collaborator selection achieved Dice scores of 0.74, 0.73, and 0.68 for Enhancing Tumor, Tumor Core, and Whole Tumor segmentation, respectively, after only 10 communication rounds. Our findings indicate that Bayesian SimAgg achieves convergence approximately 2 times faster than the non-Bayesian version. It is effective in addressing the challenges of FL, offering a promising framework for advancing federated tumor segmentation.