Much evidence suggests that both monocular and binocular depth are also determined empirically, placing on the same probabilistic footing two aspects of vision that, historically, have been regarded as having different physiological bases.

The phenomenology of distance perception clearly provides some puzzles that need to be explained. As illustrated in Figure 1, it has long been known that the apparent distance of objects bears a peculiar relationship to their physical distance from the observer. When subjects are asked to make judgments with little or no contextual information (e.g., the distance of a luminous but otherwise featureless object in a darkened room), the distances reported differ in several ways from the corresponding physical distances. First, objects in these circumstances are typically perceived to be at a distance of 2-4m, a phenomenon referred to as the "specific distance tendency" (Figure 1A). Second, objects that are relatively near each other in the retinal image appear to be about the same distance from the observer, a phenomenon called the "equidistance tendency" (Figure 1B). Third, when presented at or near eye-level, the distance of objects relatively near the observer tends to be overestimated, whereas the distance of objects that are further away tends to be underestimated (Figure 1C). Finally, the apparent distance of objects on the ground varies with the angle of declination of the line of sight: objects on the ground that are at least several meters away appear closer than they really are, and with increasing distance are judged to be progressively more elevated than warranted by their physical position (Figure 1D).

Although a variety of explanations have been proposed in the various studies cited, there has been little or no agreement about the basis of these unusual perceptions of egocentric distance. More often than not, the several tendencies illustrated in Figure 1 have simply been accepted as empirical facts that are then used to rationalize other aspects of visual space.

Given the ability of the probabilistic relationship between retinal images and sources to explain a variety of other geometrical percepts, it makes sense to ask whether the probability distributions of the possible sources of visual stimuli also determine apparent distance. Using the same database of natural scene geometry described in the sections on other geometrical illusions, the anomalous perceptions of distance illustrated in Figure 1 can all be accounted for by the probability distributions of the physical distances of object surfaces from human observers (see Yang and Purves, 2003).

The ability to explain these anomalies in the perception of distance based on the statistics of the physical distances of object surfaces from the observer in natural scenes offers further evidence that rationalizing perceived geometry in a probabilistic framework is a powerful way of understanding visual space. In addition to successfully explaining the specific anomalies that have been difficult to rationalize in other ways (see Figure 1), these observations imply that the sense of egocentric distance is another manifestation of the probabilistic strategy that allows the human visual system to contend with the inherent ambiguity of visual stimuli.