Although we have been treating
P0(
x) as an independent function, according to
Equation 13 in
,
P0(
x) is related to the frequency distribution of saccades
f(
x) and probabilities for making saccades
αn(
x). Indeed, for large
x,
Equation 13 implies that
f(
x) is simply proportional to the product
P0(
x)
α0(
x). Because
f(
x) is a constant for large
x (
Equation 3), our assumption that
P0(
x) is independent of
x at large eccentricities is equivalent to assuming that
α0(
x), the probability of making a primary saccade to an attended object, is a constant at large eccentricities. Because
α0 drops out of
Equation 1, the resulting probability distribution
P(
x) is independent of that constant. To get
P(
x) to fit the data
D(
x), we could instead assume that
P0(
x) is proportional to the cone density at large eccentricities. We would then have
α0(
x) ∝ 1/
D(
x), also at large eccentricities so that the resulting
f(
x) is constant there. Our hypothesis that the distribution of cones maximizes information then predicts the form of both the initial distribution of visual objects before saccades (
P0(
x)) and the probability of making at least one saccade to an attended object(
α0(
x)) at large
x. However, these predictions are sensitive to the form of
f(
x) at large
x, which we have estimated from very limited data. At smaller eccentricities,
P(
x) depends on all the functions
αn(
x), which is why we had to ignore them and use the experimentally observable
f(
x).