Consistency and efficiency of Bayesian estimators in generalised linear inverse problems

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. Some key words: Ky Fan metric, rates of convergence, SPECT, inverse problems, Bayesian inference, nonregular likelihood, tomography. School of Mathematics, University of Edinburgh, The King’s Buildings, Edinburgh EH9 3JZ, UK. Email: [email protected] School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. Email: [email protected].