Adaptive or dynamic signal sampling in sensing systems can adapt subsequent sampling strategies based on acquired signals, thereby potentially improving image quality and speed. This paper proposes a Bayesian method for adaptive sampling based on greedy variance reduction and stochastic gradient Langevin dynamics (SGLD). The image priors involved can be either analytical or neural network-based. Notably, the learned image priors generalize well to out-of-distribution test cases that have different statistics than the training dataset. As a real-world validation, the method is applied to accelerate the acquisition of magnetic resonance imaging (MRI). Compared to non-adaptive sampling, the proposed method effectively improved the image quality by 2-3 dB in PSNR, and improved the restoration of subtle details.