There is nothing really wrong with probability notation. As inferential steps between concepts increase in math, abuse of notation becomes indispensable.
All that probability shorthand can be unambiguously translated to formal definitions quite easily. But doing so would be analogous to writing a complex program in assembly - doable (and defined pretty much by the very fact that this is doable) but not very productive (and thus not worth doing unless you are debugging or something).
> All that probability shorthand can be unambiguously translated to formal definitions quite easily. But doing so would be analogous to writing a complex program in assembly - doable (and defined pretty much by the very fact that this is doable) but not very productive (and thus not worth doing unless you are debugging or something).
Actually I kind of disagree here.
With R or Haskell you can easily work directly with probability densities learned from data. One frequently uses the exact Bayes' rule expression with P(X), P(Y), and P(X|Y) all being known functions to get P(Y|X).
See for example functions like ecdf, which takes in an N vector of points on a 1D line and returns an actual function, namely the empirical cumulative density.
Can be very handy when you want empirical quantiles (e.g. "what percentage of the time do I expect to see 12000 hits in a day, given this single column with the hits for each of the last 200 days").
> All that probability shorthand can be unambiguously translated to formal definitions quite easily. But doing so would be analogous to writing a complex program in assembly
One possible interpretation (probably, in retrospect, the right one) is that he meant that Whitehead/Russell style axiomatization of probability was in theory possible, but would not be of much value.
I read it initially (likely wrongly in retrospect) as saying that translating the equations into an unambiguous formal computer readable definition would be intractable and/or only of theoretical interest.
All that probability shorthand can be unambiguously translated to formal definitions quite easily. But doing so would be analogous to writing a complex program in assembly - doable (and defined pretty much by the very fact that this is doable) but not very productive (and thus not worth doing unless you are debugging or something).