  Formulating a statistical inverse problem as one of inference in a Bayesian
model has great appeal, notably for what this brings in terms of coherence, the
interpretability of regularisation penalties, the integration of all
uncertainties, and the principled way in which the set-up can be elaborated to
encompass broader features of the context, such as measurement error, indirect
observation, etc. The Bayesian formulation comes close to the way that most
scientists intuitively regard the inferential task, and in principle allows the
free use of subject knowledge in probabilistic model building. However, in some
problems where the solution is not unique, for example in ill-posed inverse
problems, it is important to understand the relationship between the chosen
Bayesian model and the resulting solution.
  Taking emission tomography as a canonical example for study, we present
results about consistency of the posterior distribution of the reconstruction,
and a general method to study convergence of posterior distributions. To study
efficiency of Bayesian inference for ill-posed linear inverse problems with
constraint, we prove a version of the Bernstein-von Mises theorem for
nonregular Bayesian models.
