On each trial, the optimal observer computes the posterior probability that the target is present given the measurements, denoted
p(
C = 1|
x), and reports “target present” if this probability is greater than 0.5. This is equivalent to reporting “target present” when the log posterior ratio, denoted
d, is positive. To compute the log posterior ratio, we first apply Bayes' rule:
where
ppresent is the observer's prior probability that the target is present. (This does not have to be equal to 0.5, the true frequency of target presence.) The likelihood function of
C,
p(
x|
C), is computed by marginalizing over both
s and
T = (
T1, … ,
TN). After some basic math, we find
where
di is defined as
The relationship between
d and
di in
Equation 2 would be different if distractor orientations were not drawn independently or if more than a single target could be present. When distractors are heterogeneous and drawn from a uniform distribution, the distractor distribution is
p(
si|
Ti = 0) = 1/(2
π). Using this expression as well as
Equation 1,
Equation 3 becomes (Ma et al.,
2011)
When distractors are homogeneous with an orientation equal to
sD, we use Gaussian distributions, and
Equation 3 becomes (Peterson et al.,
1954)
We obtained the predictions of the model for an individual trial by drawing 10,000 sets of
N measurements each from Von Mises (or Gaussian) distributions centered on the respective stimuli on that trial and applying the decision rule to each set of measurements. This results in a predicted probability that the subject will report “target present” on that trial,
, where
θ denotes the model parameters.