Spectral gaps and error estimates for infinite-dimensional Metropolis-Hastings with non-Gaussian priors

We study a class of Metropolis-Hastings algorithms for target measures that are absolutely continuous with respect to a large class of non-Gaussian prior measures on Banach spaces. The algorithm is shown to have a spectral gap in a Wasserstein-like semimetric weighted by a Lyapunov function. A number of error bounds are given for computationally tractable approximations of the algorithm including bounds on the closeness of Ces\'{a}ro averages and other pathwise quantities via perturbation theory. Several applications illustrate the breadth of problems to which the results apply such as discretization by Galerkin-type projections and approximate simulation of the proposal.