The ideal-observer model is adapted from that of Najemnik and Geisler (
2005). In our adaptation, fully described in the
Supplement, we modify the model to reflect the differential payoffs for correct target localizations. Consider a search task in which the precise form of the target is known to the observer and no eye movements are made. The ideal observer proceeds by means of a template match (Green & Swets,
1966), i.e., cross correlation of the retinal image with a representation of the expected target. Given a template response at each potential target location,
i, the ideal observer computes the posterior probability of the target occurring at each location by Bayes' rule (Coombs, Dawes, & Tversky,
1970). In our case (see
Supplement for derivation), this simplifies to
where
n is the number of potential target locations,
p(
i) is the prior probability of the target occurring at position
i, and
w1 = (
w1,1, ···,
wn,1) is the vector of noisy template responses from each potential target location collected during the the first fixation (
F1, the central fixation), each with corresponding detectability value
Display Formula based on the visibility map described earlier. In our experiment,
n = 8, and
p(
i) = 1/8, and a reward
Vi is awarded for correctly detecting the target at each position
i. The ideal observer selects the target position,
I, with maximal expected gain: