Abstract The ice shelves buttressing the Antarctic ice sheet determine the rate of ice-discharge into the surrounding oceans. Their geometry and buttressing strength are influenced by the local surface accumulation and basal melt rates, governed by atmospheric and oceanic conditions. Contemporary methods quantify one of these rates, but typically not both. Moreover, information about these rates is only available for recent time periods, reaching at most a few decades back since measurements are available. We present a new method to simultaneously infer the surface accumulation and basal melt rates averaged over decadal and centennial timescales. We infer the spatial dependence of these rates along flow line transects using internal stratigraphy observed by radars, using a kinematic forward model of internal stratigraphy. We solve the inverse problem using simulation-based inference (SBI). SBI performs Bayesian inference by training neural networks on simulations of the forward model to approximate the posterior distribution, therefore also quantifying uncertainties over the inferred parameters. We validate our method on a synthetic example, and apply it to Ekström Ice Shelf, Antarctica, for which independent validation data are available. We obtain posterior distributions of surface accumulation and basal melt averaging over up to 200 years before 2022.