Efficient Bayesian Inference in Strictly Semi-parametric Linear Inverse Problems

We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal $f$ corrupted by statistical noise, with the transformation $K_\theta$ being linear but unknown up to a scalar $\theta$. We adopt a Bayesian approach and put a prior on the pair $(\theta,f)$ and prove a Bernstein-von Mises theorem for the marginal posterior of $\theta$ under regularity conditions on the operators $K_\theta$ and on the prior. We apply our results to the recovery of location parameters in semi-blind deconvolution problems and to the recovery of attenuation constants in X-ray tomography.