I don't think that the author's points are valid. He seems to be using references with sloppy notation.
Suffice it to say that if he picks up a mathematical probability textbook he should be satisfied.
* I do agree that people use shortcuts to make equations seem simpler, that some standard equations look complicated, and that you need to think hard about which scenario is appropriate for your application.
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.
Suffice it to say that if he picks up a mathematical probability textbook he should be satisfied.
* I do agree that people use shortcuts to make equations seem simpler, that some standard equations look complicated, and that you need to think hard about which scenario is appropriate for your application.