How to combine direction cues optimally

Abstract

Many perceptual tasks require estimating a direction in space. Often several directional cues are available, such visual and gravitational cues to the subjective vertical, or visual and auditory cues to the direction of an object. In work on the subjective vertical, researchers have developed a heuristic vector summation model that has no deep theoretical motivation, but that accounts well for the direction and reliability of observers’ direction estimates when multiple cues are available, and that can accommodate directional cues ranging over all possible directions (Mittelstaedt, 1983). In work on combining visual and auditory cues to direction, researchers have used statistically motivated cue combination models that were originally developed for linear quantities such as depth, not circular or spherical quantities such as direction, and hence work only over a limited range of cue directions (Alais and Burr, 2004). Here we present a new model of directional cue combination that combines the advantages of both previous approaches. We develop a statistical theory of cue combination based on the von Mises distribution, the analog on the circle of the normal distribution on the line. We show that this theory differs in important ways from the theory of linear cue combination, e.g., a combined direction estimate can be less certain than any of the individual cues that were used to calculate it. We also show that the vector summation model developed empirically by previous investigators is an excellent approximation to our theory, meaning that it is a nearly optimal way of combining directional cues.