Towards exact MPM-decision for Labelling Problems Torsten Wierschin Technische Universitat Dresden Generalization of the usual maximum a posteriori decision (MAP) on random fields is presented. Although considerable attention has been paid to prior modeling in Bayes-imaging, little work has gone to finding suitable estimators through designing loss functions. Loss functions with a local structure, where the decision at one node (of the labelled graph) depends on the pending decisions in the neighborhood are suggested. It is shown, that there exist a wide family of such. Here, the MAP-function plays the role of totally ignoring local infomation. Its opposite the so called maximum-posterioir-marginal decision (MPM) is introduced. To compute the MPM estimate on an MRF, we use exact sampling from Gibbs distributions based on a recently developed technique called Coupling from the Past for Markov Chains. Additionally, we present the first (?) exact computation of the MPM decision for the m-Line, the m-Cycle and the m-Ladder in polynomial time of m. Imaging result for a stereo task is given.