PC-MRI based 4D flow imaging is a powerful tool to quantify hemodynamics within the heart and the great vessels. We develop a Bayesian technique to greatly accelerate 4D flow image acquisition. Our technique exploits the rich structure in 4D flow images within a joint reconstruction algorithm. We validate the technique using retrospectively accelerated flow phantom data and prospectively accelerated, single breath-hold in vivo data. In the flow phantom, stroke volume differed by ≤9% for R≤28 when compared to fully sampled data. The high acceleration provided by the Bayesian approach could allow for clinical application of single breath-hold 4D flow imaging.