We must establish a common denominator for comparison of position estimates given measurements. We have pursued a statistical procedure for doing so. Our objective is to maximize the posterior probability of position given the bearings, . Position is discretized in intervals of 30 meters, agreeing with the discretization of Digital Elevation Maps. At first, we must specify a prior density for position, . Currently we use a uniform distributions to signify absence of prior knowledge. Secondly, we must specify the likelihood the bearings (the measurements) given position, . In practice it is difficult to determine the likelihood of a bearing given position; it is appropriate to decompose this expression in terms of the possible interpretations of the bearings:The final posterior density is proportional to .
Experience with the mountain detection system has led to the conclusion that, for fixed pose and correspondences, the errors in measuring bearings are fairly independent:We assume that when a landmark is visible from , the errors in the bearing follow a Gaussian law. We also measure the height of the mountain associated with the bearing, and assume errors in height follow a Gaussian law. We consider that any error that exceeds 18 degrees is a mistake, that is, if a mountain at 40 degrees is associated to a bearing at 60 degrees, we consider that association a mistake. Mistakes are assumed distributed uniformly on the interval [0, 360] degrees. Every bearing is associated with all mountains within a degrees interval; bearings that are not associated to any mountain are mistakes. Since bearings do not need to be associated to all possible mountains, the summation in expression 1 can be calculated quickly.