  This paper analyzes a popular computational framework to solve
infinite-dimensional Bayesian inverse problems, discretizing the prior and the
forward model in a finite-dimensional weighted inner product space. We
demonstrate the benefit of working on a weighted space by establishing
operator-norm bounds for finite element and graph-based discretizations of
Mat\'ern-type priors and deconvolution forward models. For linear-Gaussian
inverse problems, we develop a general theory to characterize the error in the
approximation to the posterior. We also embed the computational framework into
ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our
operator-norm bounds for prior discretizations guarantee the scalability and
accuracy of these algorithms under mesh refinement.
