Exact MAP states and expectations from perfect sampling: Greig, porteous and seheult revisited

In 1989 Greig, Porteous and Seheult showed that the maximum a posteriori (MAP) state can be exactly calculated for degraded binary images. Their interest was in assessing the performance of algorithms used to find the MAP state, such as simulated annealing. A secondary conclusion was that the MAP state, at least in the restricted setting of two-color images, does not provide a robust reconstruction of the true image. That result has been interpreted by some as indicating that the Ising MRF used by GPS is not a good prior model for such images. We show that such a judgement is premature as the MAP state does not well summarize the information in the posterior distribution in this case. In particular, the deviation of the MAP state from the mean, particularly at larger smoothing parameters, shows that the MAP state is not representative of the bulk of feasible reconstructions. We calculate other summary statistics that interpret and display the information in the posterior by implementing full Bayesian infe...