A method is presented for the analysis of dynamic positron emission tomography (PET) data using sparse Bayesian learning. Parameters are estimated in a compartmental framework using an over-complete exponential basis set and sparse Bayesian learning. The technique is applicable to analyses requiring either a plasma or reference tissue input function and produces estimates of the system's macro-parameters and model order. In addition, the Bayesian approach returns the posterior distribution which allows for some characterisation of the error component. The method is applied to the estimation of parametric images of neuroreceptor radioligand studies.