I upvoted because distillations are important, but a first pass over your post left me much more confused than I was before. Another issue was the level of formality. Either use less, and stick to core intuitions, or use more. Personally, I would have liked to know the types of all the objects being mentioned, and a mention of what space X was (I’m assuming the space of possible events?), and some explanation of how PHg relates to P+/P− or even g. In fact, I can’t see how g relates to any of the objects.
EDIT: I meant to say, you didn’t say how g relates to any of the distributions or update rules or whatever except via an unclear analogy. Though note, I’m pretty tired right now so take my confusion as not indicative of the average reader.
Thank you for your comment! You are right, these things are not clear from this post at all and I did not do a good job at clarifying that. I’m a bit low on time atm, but hopefully, I’ll be able to make some edits to the post to set the expectations for the reader more carefully.
The short answer to your question is: Yep, X is the space of events. In Vanessa’s post it has to be compact and metric, I’m simplifying this to an interval in R. And P+/P− can be derived from PHg by plugging in g=0 and replacing the measure m(A) by the Lesbegue integral ∫Adm. I have scattered notes where I derive the equations in this post. But it was clear to me that if I want to do this rigorously in the post, then I’d have to introduce an annoying amount of measure theory and the post would turn into a slog. So I decided to do things hand-wavy, but went a bit too hard in that direction.
I upvoted because distillations are important, but a first pass over your post left me much more confused than I was before. Another issue was the level of formality. Either use less, and stick to core intuitions, or use more. Personally, I would have liked to know the types of all the objects being mentioned, and a mention of what space X was (I’m assuming the space of possible events?), and some explanation of how PHg relates to P+/P− or even g. In fact, I can’t see how g relates to any of the objects.
EDIT: I meant to say, you didn’t say how g relates to any of the distributions or update rules or whatever except via an unclear analogy. Though note, I’m pretty tired right now so take my confusion as not indicative of the average reader.
Thank you for your comment! You are right, these things are not clear from this post at all and I did not do a good job at clarifying that. I’m a bit low on time atm, but hopefully, I’ll be able to make some edits to the post to set the expectations for the reader more carefully.
The short answer to your question is: Yep, X is the space of events. In Vanessa’s post it has to be compact and metric, I’m simplifying this to an interval in R. And P+/P− can be derived from PHg by plugging in g=0 and replacing the measure m(A) by the Lesbegue integral ∫Adm. I have scattered notes where I derive the equations in this post. But it was clear to me that if I want to do this rigorously in the post, then I’d have to introduce an annoying amount of measure theory and the post would turn into a slog. So I decided to do things hand-wavy, but went a bit too hard in that direction.