The majority of research on efficient and scalable algorithms in computational science and engineering has focused on the forward problem: given parameter inputs, solve the governing equations to determine output quantities of interest. Here we consider the broader question: given a model containing uncertain parameters, noisy observational data, and a prediction quantity of interest (QOI), how do we construct efficient and scalable algorithms to (1) infer model parameters from the data (the deterministic inverse problem), (2) quantify uncertainty in the inferred parameters (the Bayesian inference problem), and (3) propagate the resulting uncertain parameters through the model for predictions with quantified uncertainties on the QOI (the forward uncertainty propagation problem)? We present efficient and scalable algorithms for this end-to-end, data-to- prediction process in the context of modeling the flow of the Antarctic ice sheet and its effect on loss of grounded ice to the ocean. Ice is modeled as a viscous, incompressible, creeping, shear-thinning fluid, the observational data come from satellite measurements of surface ice flow velocity, and the uncertain parameter field inferred is a basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the base of the ice sheet. The QOI is the present-day ice mass flux from the Antarctic continent to the ocean. We show that the work required for executing this data-to- prediction process is independent of the state dimension, parameter dimension, data dimension, and the number of processor cores. The key to achieving this dimension independence is to exploit the fact that, despite their large size, observational data typically provide sparse information on model parameters. This property is exploited to construct a low rank approximation of the parameter-to-observable map via randomized SVD methods and adjoint-based actions of Hessians of the data misfit functional.