Title: Closed loop deep Bayesian inversion:  Uncertainty driven acquisition for fast MRI. Abstract: This work proposes a closed-loop, uncertainty-driven adaptive sampling frame- work (CLUDAS) for accelerating magnetic resonance imaging (MRI) via deep Bayesian inversion. By closed-loop, we mean that our samples adapt in real- time to the incoming data. To our knowledge, we demonstrate the first genera- tive adversarial network (GAN) based framework for posterior estimation over a continuum sampling rates of an inverse problem. We use this estimator to drive the sampling for accelerated MRI. Our numerical evidence demonstrates that the variance estimate strongly correlates with the expected MSE improvement for dif- ferent acceleration rates even with few posterior samples. Moreover, the resulting masks bring improvements to the state-of-the-art fixed and active mask designing approaches across MSE, posterior variance and SSIM on real undersampled MRI scans.