Differences between the left and right eye＇s views of the world carry information about three-dimensional scene structure and about the position of the eyes in the head The contemporary Bayesian approach to perception(1,2) implies that human performance in using this source of eye-position information can be analysed most usefully by comparison with the performance of a statistically optimal observer. Here we argue that the comparison observer should also be statistically robust, and we find that this requirement leads to qualitatively new behaviours. For example, when presented with a class of stereoscopic stimuli containing inconsistent information about eccentricity of gaze, estimates of this gaze parameter recorded from one robust ideal observer bifurcate at a critical value of stimulus inconsistency. We report an experiment in which human observers also show this phenomenon and we use the experimentally determined critical value to estimate the vertical acuity of the visual system. The Bayesian analysis also provides a highly reliable and biologically plausible algorithm that can recover eye positions even before the classic stereo-correspondence problem is solved, that is, before deciding which features in the left and right images are to be matched.