(1) Theory: There are many perceptual and other situations in which a participant has many different cues available for making a particular estimate or decision. For example, in the perception of depth, there are literally dozens of cues (pictorial, binocular, movement, etc.). Each cue is noisy. What is the optimal estimate or decision given a collection of noisy cues?

(2) Theory: If one is willing to assume that the cues individually lead to unbiased, independent estimates and one requires the use of a linear cue-combination rule leading to minimum overall variance, the resulting rule is a weighted average of the individual cue estimates with weights proportional to cue reliabilities (defined as the inverse of the variance from each cue).

(3) Theory: An alternative approach is to use Bayesian methods, combining the noisy perceptual data with a prior. In the case of Gaussian likelihoods and priors and conditionally independent cues, for several combination rules (maximum a posteriori, mean of the posterior and median) the optimal estimate uses the same rule as in (2) with the prior acting as an additional, static cue.

(4) Experiment: One can test these theories by using cue-conflict stimuli in which one cue is perturbed relative to other cues. By measuring the participant's overall estimate as a function of the perturbation applied to one cue, one can estimate the weight the observer applied to that cue (and compare it to the optimal weight given the cue variances). For many cue combination problems (e.g., combining visual cues to depth, visual and haptic cues to size, etc.), both the measured cue weights and the variance of the multi-cue estimate are roughly consistent with the predictions of the optimal cue-combination scheme.

(5) Theory: For some mutliple-cue scenarios, the observer needs to determine in addition whether the cues resulted from the same source in the world. For example, whether a sound and a visual stimulus came from the same or separate locations. If they came from the same source, then location estimates from each should be combined. If not, they should be kept separate. This leads to more complex Bayesian models that simultaneously estimate the probability of a common source and the best estimates of the location of each cue. These combination rules are not linear.

(6) Theory: The overarching theory here is Statistical Decision Theory. This theory concerns three items: states of the world, sensory data, and decisions or actions the observer makes. Pairs of these items are linked by the likelihood function (the probability of a sensory state given a particular state of the world), the decision function (the action taken by the observer given a particular sensory state) and the cost function (the consequences for the participant given a particular decision and state of the world). The optimal decision function depends on how the participant ranks different possible outcomes. One optimality principle is to use Bayesian decision theory for which an additional item is needed: the prior probability of states of the world. The optimal Bayesian decision function is one that minimizes expected risk. We can ask whether human subjects behave like optimal Bayesian decision makers by manipulating the cost function experimentally.