We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with different levels of uncertainty in the operator. Using product priors, we prove contraction rates for the posterior distribution which coincide with the optimal convergence rates up to logarithmic factors. In order to adapt to the unknown smoothness, an empirical Bayes procedure is constructed based on Lepski's method. The procedure is illustrated in numerical examples.