Bayesian inverse problems with non-conjugate priors

We investigate the frequentist posterior contractionrate of nonparametric Bayesian procedures in linear inverse problems in both the Mildly and severely ill-posed cases. A theorem is provedin a general Hilbert Space setting under approximation-theoretic assumptions on the prior. The Resultis applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, over smoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform.