= 27d567bc2d48d9f40e9ccc20ea370b15)
Kaj_Sotala comments on Open Thread, April 1-15, 2012 - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (150)
I am currently considering the question "Does probability have a smallest divisible unit?" and I think I'm confused.
For instance, it seems like time has a smallest divisible unit, Planck time. Whereas the real numbers do not have a smallest divisible unit. Instead, they have a Dense order. So it seems reasonable to ask "Does probability have a smallest divisible unit?"
Then to try answering the question, if you describe a series of events which can only happen in 1 particular branch of the many worlds interpretation, and you describe something which happens in 0 branches of the many worlds interpretation, then my understanding is there is no series of events which has a probability in between those two things, which would appear to imply the concept of a smallest unit of probability is coherent and the answer is "Yes."
However, there is an article on Infinitely divisible probability and if you can divide something infinitely, then of course, the concept of it having a smallest unit is nonsensical, and the answer would be "No."
How do I resolve this confusion?
What would you anticipate to be different if probability did/didn't have a smallest divisible unit?
Pascal's wager, for one thing.
How's this? (I'm thinking here that the smallest unit would correspond to 1 possible arrangement of the Hubble volume, so the unit would be something like 1/10^70 or something. Any other state of the world is meaningless since it can't exist.)
As usually formulated, Bayesian probability maps beliefs onto the reals between 0 and 1, and so there's no smallest or largest probability. If you act as if there is and violate Cox's theorem, you ought to be Dutch bookable through some set of bets that either split up extremely finely events (eg. a dice with trillions of sides) or aggregated many events. If there is a smallest physical probability, then these Dutch books would be expressible but not implementable (imagine the universe has 10^70 atoms - we can still discuss 'what if the universe had 10^71 atoms?').
This leads to the observed fact that an agent implementing probability with units is Dutch bookable in theory, but you will never observe you or another agent Dutch booking said agent. It's probably also more computationally efficient.
Good answer to help me focus.
If probability has a smallest divisible unit, it seems like there would have to be one or more least probable series of events.
If I was to anticipate that there was one or more least probable series of events, it seems like I would have to also anticipate that additional events will stop occurring in the future. If events are still taking place, a particular even more complicated series of events can continue growing more improbable than whatever I had previously thought of as a least probable event.
So it seems an alternative way of looking at this question is "Do I expect events to still be taking place in the future?" In which case I anticipate the answer is "Yes" (I have no evidence to suggest they will stop) and I think I have dissolved the more confusing question I was starting with.
Given that that makes sense to me, I think my next step is if it makes sense to other people. If I've come up with an explanation which makes sense only to me, that doesn't seem likely to be helpful overall.
Makes sense to me.