Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations

We obtain rates of contraction of posterior distributions in inverse problems with discrete observations. In a general setting of smoothness scales we derive abstract results for general priors, with contraction rates determined by discrete Galerkin approximation. The rate depends on the amount of prior concentration near the true function...

The Doctrine of Bayesian Statistics for Inverse Problems

Imagine you need to navigate through a completely dark cave. A well-known way of achieving this is echolocation, which works by making sounds and listening to how they are reflected back. The problem of determining the geometric shape of a space from a mixture of reflections of emitted sounds, is an example of an inverse problem. In many fields...

A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density f0 of its jump sizes, as well as of its intensity λ0. We take a Bayesian approach to the problem and specify the prior on f0 as the Dirichlet location mixture of normal densities. An independent prior for λ0 is assumed to be...