Yeah, also note that the history of X given Y is not actually a well defined concept. There is only the history of X given y for y∈Y. You could define it to be the union of all of those, but that would not actually be used in the definition of orthogonality. In this case hF(X|y), hF(V|y), and hF(Z|y) are all independent of choice of y∈Y, but in general, you should be careful about that.
Yeah, fair point. (I did get this right in the summary; turns out if you try to explain things from first principles it becomes blindingly obvious what you should and shouldn’t be doing.)
I think that works, I didn’t look very hard. Yore histories of X given Y and V given Y are wrong, but it doesn’t change the conclusion.
Yeah, both of those should be {X,V}, if I’m not mistaken (a second time).
Yeah, also note that the history of X given Y is not actually a well defined concept. There is only the history of X given y for y∈Y. You could define it to be the union of all of those, but that would not actually be used in the definition of orthogonality. In this case hF(X|y), hF(V|y), and hF(Z|y) are all independent of choice of y∈Y, but in general, you should be careful about that.
Yeah, fair point. (I did get this right in the summary; turns out if you try to explain things from first principles it becomes blindingly obvious what you should and shouldn’t be doing.)