Semi-parametric Bernstein-von Mises theorem in linear inverse problems

We consider a Bayesian approach for the recovery of scalar parameters arising in inverse problems. We consider a general signal-in white noise model where we have access to two independent noisy observations of a function, and of a linear transformation of the function. The linear operator is unknown up to a scalar parameter. We present a Bernstein-von Mises theorem for the marginal posterior of the scalar under regularity assumptions of the operator. We further derive Bernstein-von Mises results for different priors and apply them to two concrete examples: the recovery of the thermal diffusivity in a heat equation problem, and the recovery of a location parameter in a semi-blind deconvolution problem.