Here I’m going to log some things I notice while reading, mostly as a way to help check my understanding, but also to help find possible errata.
In Definition part (a), you’ve got a whole lot of W-type symbols, and I’m not 100% sure I follow each of their uses. You use ω a couple times which is legit, but it looks a lot like w, so maybe it could be replaced with N?
See this comment for two errata with the different w’s.
Fut(w) denotes, for any world state w∈W , the future of the Dirac (100% concentrated) distribution on the world state w∈W.
Maybe you could just say, Fut(w) is shorthand for Fut(TW(w)), since TW will map w to the right thing of type ΔW. Then you can avoid bringing in the somewhat exotic Dirac delta function. Of course, that now means that w itself is not the first item in the resulting sequence. I’m not sure if you need that to be the case for later. But also, everything above is ambiguous about whether the argument to Fut was in the sequence anyway.
The character ⫫ doesn’t render for me. (I could figure out what it was by pasting the unicode into google, but maybe it could be done with LaTeX instead?)
To formalize this, I want a collection of state spaces and maps, like so:
Is the following bulleted list missing an entry for E?
Each of these factorizations are assumed to be bijective, in the sense of accounting for everything that matters and not double-counting anything
I was wondering if you were going to say something like W=V×B×E and B=A×P. It sounds like that’s almost right, except that you allow the factors to pass through arbitrary functions first, as long as they’re bijective. Is that right?
We say rθ is a good fit
You bring back rθ here, but I don’t see the θ doing anything yet. Might be better not to introduce it until later, to free up a bit of the reader’s working memory.
Here I’m going to log some things I notice while reading, mostly as a way to help check my understanding, but also to help find possible errata.
In Definition part (a), you’ve got a whole lot of W-type symbols, and I’m not 100% sure I follow each of their uses. You use ω a couple times which is legit, but it looks a lot like w, so maybe it could be replaced with N?
See this comment for two errata with the different w’s.
Maybe you could just say, Fut(w) is shorthand for Fut(TW(w)), since TW will map w to the right thing of type ΔW. Then you can avoid bringing in the somewhat exotic Dirac delta function. Of course, that now means that w itself is not the first item in the resulting sequence. I’m not sure if you need that to be the case for later. But also, everything above is ambiguous about whether the argument to Fut was in the sequence anyway.
The character ⫫ doesn’t render for me. (I could figure out what it was by pasting the unicode into google, but maybe it could be done with LaTeX instead?)
Is the following bulleted list missing an entry for E?
I was wondering if you were going to say something like W=V×B×E and B=A×P. It sounds like that’s almost right, except that you allow the factors to pass through arbitrary functions first, as long as they’re bijective. Is that right?
You bring back rθ here, but I don’t see the θ doing anything yet. Might be better not to introduce it until later, to free up a bit of the reader’s working memory.
See this comment for a broken link.