Looking again at Theorem 4.4, I don’t find any assumption analogous to the independence property. If (D2,μ2)=(D1,μ1), won’t this wrongly claim that the values of the optimal predictor should be squared along the diagonal?
There is a crucial difference between the setting of Theorem 4.4 and setting of Theorems 4.5, 4.6. In Theorems 4.5, 4.6 we consider the intersection of two languages in which case D∩D=D. In Theorem 4.4 we consider the Cartesian product of two languages.(D×D,μ×μ) is not the same thing as (D,μ). Moreover, the diagonal embedding of (D,μ) into (D×D,μ×μ) is not a valid reduction for the purpose of Theorem 6.1 since condition (ii) is violated (diagonal elements are rare within the product ensemble).
Looking again at Theorem 4.4, I don’t find any assumption analogous to the independence property. If (D2,μ2)=(D1,μ1), won’t this wrongly claim that the values of the optimal predictor should be squared along the diagonal?
(Apologies if I’m misunderstanding.)
There is a crucial difference between the setting of Theorem 4.4 and setting of Theorems 4.5, 4.6. In Theorems 4.5, 4.6 we consider the intersection of two languages in which case D∩D=D. In Theorem 4.4 we consider the Cartesian product of two languages.(D×D,μ×μ) is not the same thing as (D,μ). Moreover, the diagonal embedding of (D,μ) into (D×D,μ×μ) is not a valid reduction for the purpose of Theorem 6.1 since condition (ii) is violated (diagonal elements are rare within the product ensemble).