  In this article, we study Bayesian inverse problems with multi-layered
Gaussian priors. We first describe the conditionally Gaussian layers in terms
of a system of stochastic partial differential equations. We build the
computational inference method using a finite-dimensional Galerkin method. We
show that the proposed approximation has a convergence-in-probability property
to the solution of the original multi-layered model. We then carry out Bayesian
inference using the preconditioned Crank--Nicolson algorithm which is modified
to work with multi-layered Gaussian fields. We show via numerical experiments
in signal deconvolution and computerized X-ray tomography problems that the
proposed method can offer both smoothing and edge preservation at the same
time.
