  We characterize the measurement complexity of compressed sensing of signals
drawn from a known prior distribution, even when the support of the prior is
the entire space (rather than, say, sparse vectors). We show for Gaussian
measurements and \emph{any} prior distribution on the signal, that the
posterior sampling estimator achieves near-optimal recovery guarantees.
Moreover, this result is robust to model mismatch, as long as the distribution
estimate (e.g., from an invertible generative model) is close to the true
distribution in Wasserstein distance. We implement the posterior sampling
estimator for deep generative priors using Langevin dynamics, and empirically
find that it produces accurate estimates with more diversity than MAP.
