Bayesian combination of two-dimensional location estimates

We extend a Bayesian method for combining estimates of means and variances from independent cues in a spatial cue-combination paradigm. In a typical cue-combination experiment, subjects estimate a value on a single dimension—for example, depth—on the basis of two different cues, such as retinal disparity and motion. The mathematics for this one-dimensional case is well established. When the variable to be estimated has two dimensions, such as location (which has both x and y values), then the one-dimensional method may be inappropriate due to possible correlations between x and y and the fact that the dimensions may be inseparable. A cue-combination task for location involves people or animals estimating xy locations under two single-cue conditions and in a condition in which both cues are combined. We present the mathematics for the two-dimensional case in an analogous manner to the one-dimensional case and illustrate them using a numeric example. Our example involves locations on maps, but the method illustrated is relevant for any task for which the estimated variable has two or more dimensions.