Just a quick rebuttal of the author's specific points:
(1) Given a set of elements X, E(X) = \sum{x \in X} xp(x). The problem the author mentioned is solved, since we are now summing over all elements in X rather than using the input variable inappropriately.
(2) Given sets of elements X and Y, and the set of ALL elements O, then p(X), p(Y), and p(X|Y) are all computed in the same manner. p(X) is shorthand for p(X|O) -- so we are now given three analogous functions, p(X|O), p(Y|O), and p(X|Y). So, Bayes' can be used to compute all three in the exact same manner, if you so wish.
The above rebuttals are obviously discrete, but there are analogous continuous variable scenarios.
(1) Given a set of elements X, E(X) = \sum{x \in X} xp(x). The problem the author mentioned is solved, since we are now summing over all elements in X rather than using the input variable inappropriately.
(2) Given sets of elements X and Y, and the set of ALL elements O, then p(X), p(Y), and p(X|Y) are all computed in the same manner. p(X) is shorthand for p(X|O) -- so we are now given three analogous functions, p(X|O), p(Y|O), and p(X|Y). So, Bayes' can be used to compute all three in the exact same manner, if you so wish.
The above rebuttals are obviously discrete, but there are analogous continuous variable scenarios.