Communicated by I. SevostianovWe pose a Bayesian formulation of the inverse problem associated to recovering both the support and the refractive index of a convex obstacle given measurements of near-field scattered waves. Aiming at sampling efficiently from the arising posterior distribution using Markov Chain Monte Carlo, we construct a sampler (probability transition kernel) that is invariant under affine transformations of space. A point cloud method is used to approximate the scatterer support. We show that affine invariant sampling can successfully address the presence of multiple scales in inverse scattering in inhomogeneous media.In this section, we formulate the forward problem in terms of the Lippmann-Schwinger equation and discuss its discretization through the method of Vainikko [14]. Next, we present the Bayesian approach to inverse scattering problems, and we show the construction of The example involves an elliptical inclusion centered at .0, 0/ with principal axes 0.3 and 0.05 and with a contrast ratio of 25 : 1 in the refractive index, that is, b D 25. We fix the instrumental parameters in the prior probability distribution .b/ of the refractive