Cue combination on the circle and the sphere

Bayesian cue combination models have been used to examine how human observers combine information from several cues to form estimates of linear quantities like depth. Here we develop an analogous theory for circular quantities like planar direction. The circular theory is broadly similar to the linear theory but differs in significant ways. First, in the circular theory the combined estimate is a nonlinear function of the individual cue estimates. Second, in the circular theory the mean of the combined estimate is affected not only by the means of individual cues and the weights assigned to individual cues but also by the variability of individual cues. Third, in the circular theory the combined estimate can be less certain than the individual estimates, if the individual estimates disagree with one another. Fourth, the circular theory does not have some of the closedform expressions available in the linear theory, so data analysis requires numerical methods. We describe a vector summodel that gives a heuristic approximation to the circular theory's behavior. We also show how the theory can be extended to deal with spherical quantities like direction in three-dimensional space.