  We solve the inverse problem of deblurring a pixelized image of Jupiter using
regularized deconvolution and by sample-based Bayesian inference. By
efficiently sampling the marginal posterior distribution for hyperparameters,
then the full conditional for the deblurred image, we find that we can evaluate
the posterior mean faster than regularized inversion, when selection of the
regularizing parameter is considered. To our knowledge, this is the first
demonstration of sampling and inference that takes less compute time than
regularized inversion in an inverse problems. Comparison to random-walk
Metropolis-Hastings and block Gibbs MCMC shows that marginal then conditional
sampling also outperforms these more common sampling algorithms, having better
scaling with problem size. When problem-specific computations are feasible the
asymptotic cost of an independent sample is one linear solve, implying that
sample-based Bayesian inference may be performed directly over function spaces,
when that limit exists.
