  We investigate the frequentist posterior contraction rate of nonparametric
Bayesian procedures in linear inverse problems in both the mildly and severely
ill-posed cases. A theorem is proved in a general Hilbert space setting under
approximation-theoretic assumptions on the prior. The result is 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, oversmoothing 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.
