The quantification of myocardial perfusion MRI has the potential to provide a fast, automated and user-independent assessment of myocardial ischaemia. However, due to the relatively high noise level and low temporal resolution of the acquired data and the complexity of the tracer-kinetic models, the model fitting can yield unreliable parameter estimates. A solution to this problem is the use of Bayesian inference which can incorporate prior knowledge and improve the reliability of the parameter estimation. This, however, uses Markov chain Monte Carlo sampling to approximate the posterior distribution of the kinetic parameters which is extremely time intensive. This work proposes training convolutional networks to directly predict the kinetic parameters from the signal-intensity curves that are trained using estimates obtained from the Bayesian inference. This allows fast estimation of the kinetic parameters with a similar performance to the Bayesian inference.