Bernstein-von Mises theorem for nonregular generalised linear inverse problems
Natalia Bochkina (University of Edinburgh)
Friday 15th March, 2013 15:00-16:00 Maths 204
This theoretical study is motivated by an example from single photon emission computed tomography (SPECT), a medical imaging technique that involves a tomographic reconstruction of the spatial pattern of a radioactively-labelled substance, known to concentrate in the tissue to be imaged. In the Bayesian model for this problem considered by P. Green (1990) that is an ill-posed inverse problem with Poisson likelihood and a pairwise-interaction Markov random field prior, the likelihood is not identifiable and the prior distribution is improper which raises questions about convergence and concentration of the posterior distribution for this model.
We present results on convergence of the posterior distribution and its a local approximation around the limit. For regular Bayesian models, such a result is known as the Bernstein-von Mises theorem that states that the posterior distribution is approximately Gaussian. Applied to calculating functionals of the posterior distribution such as the expected value and the variance, this technique is known as the Laplace approximation that has been shown to be a good competitor to stochastic simulation methods for performing Bayesian inference. It turns out, that the Bayesian model for the SPECT example is non-regular in two ways: the likelihood is not identifiable and the intensities of the true image lie on the boundary of the parameter space. We show how this affects the rates of convergence of the posterior distribution, and that the local approximation of the posterior distribution is not always Gaussian.
This is joint work with Peter Green (University of Bristol).