Quote:
Originally Posted by marek
I am still confused on how to piece everything together. I like your much cleaner version, but I do have one comment. Having disjoint hypothesis sets does not necessarily mean that the set of dichotomies they create will also be disjoint.
For example, let H1 and H2 be the positive and negative 1d rays, respectively. These two hypothesis sets are disjoint. However, given any data set they can both create the dichotomies of all +1 or all 1. They won't have much overlap beyond that (and maybe THAT is the point), but they won't be entirely disjoint as far as our inequalities are concerned.

Quote:
Originally Posted by TTHotShot
Great work everyone  this question had me scratching my head until I found this post. The one thing I don't understand is the above statment. Am I missing something? It seems like it should be an inequality.

@ marek, TTHotShot: Well, I agree with your remark, I was too quick: I have not explored the link between a hypothesis set and the dichotomies it generates on a set of points well enough. As a consequence the assertion that
in general is wrong in general. Instead, it should be an inequality (≤), and probably quite a loose one, since there can be an overlap in the dichotomies generated by
and
. Additionally, the definition of
involves the determination of a maximum, and the independent search of 2 max on the RHS is a lot less constraining than one max on the LHS.