Fundamental Limits of Bayesian Inference: Order Parameters and Phase Transitions for Road Tracking
There is a growing interest in formulating vision problems in terms of Bayesian inference and, in particular, the maximum a posteriori (MAP) estimator. In this paper, we consider the special case of detecting roads from aerial images and demonstrate that analysis of this ensemble enables us to determine fundamental bounds on the performance of the MAP estimate. We demonstrate that there is a phase transition at a critical value of the order parameter; below this phase transition, it is impossible to detect the road by any algorithm. We derive closely related order parameters which determine the time and memory complexity of search and the accuracy of the solution using the n* search strategy. Our approach can be applied to other vision problems, and we briefly summarize the results when the model uses the "wrong prior". We comment on how our work relates to studies of the complexity of visual search and the critical behaviour in the computational cost of solving NP-complete problems.