Seismic analysis is a key element in successful exploration and production of natural resources. During the last decades, seismic methodology has had a significant progress with respect to both acquisition, processing and analysis. Despite all the new tec hnology, the uncertainty related to seismic analysis is still large, and even worse, the uncertainty is often not systematically assessed. In this thesis, the uncertainty aspect of seismic amplitude versus offset (AVO) in version is assessed using a Bayesian approach to inversion. The main objective is to estimate elastic material parameters with associated uncertainty from large seismic data sets, but the in versionproblem also includes estimation of seismic wavelets and the noise level. State of the art statistical methodology is applied to attack these current and crucial geophysical problems. The core part of the work is presented in four separate papers written for geophysical journals, constituting Chapter 2 through 5 in this thesis. Each of the papers is self-contained, with exception of the references which are placed in a separate bibliography chapter.