Does your principle follow from Goods? It would seem that it does. Perhaps a good way to generalise the idea would be that the EV linearly aggregates the distribution and isnt expected to change, but other aggregations like log get on average closer to their value at hypothetical certainty. For example the variance of a real parameter goes expected down.
Good shows that for every utility function for every situation, the EV of utility increases or stays the same when you gain information.
If we can construct a utility function where its utility EV always equals the the EV of propabilty assigned to the correct hypothesis, we could transfer the conclusion. That was my idea when I made the comment.
Here is that utility function: first, the agent mentally assigns a positive real number r(hi) to every hypothesis hi, such that ∑ir(hi)=1. It prefers any world where it does this to any where it doesnt. Its utility function is :
I see. I think you could also use PPI to prove Good’s theorem though. Presumably the reason it pays to get new evidence is that you should expect to assign more probability to the truth after observing new evidence?
Does your principle follow from Goods? It would seem that it does. Perhaps a good way to generalise the idea would be that the EV linearly aggregates the distribution and isnt expected to change, but other aggregations like log get on average closer to their value at hypothetical certainty. For example the variance of a real parameter goes expected down.
I didn’t take the time to check whether it did or didn’t. If you would walk me through how it does, I would appreciate it.
Good shows that for every utility function for every situation, the EV of utility increases or stays the same when you gain information.
If we can construct a utility function where its utility EV always equals the the EV of propabilty assigned to the correct hypothesis, we could transfer the conclusion. That was my idea when I made the comment.
Here is that utility function: first, the agent mentally assigns a positive real number r(hi) to every hypothesis hi, such that ∑ir(hi)=1. It prefers any world where it does this to any where it doesnt. Its utility function is :
This is the quadratic scoring rule, so r(hi)=P(hi). Then its expected utility is :
Simplifying:
I see. I think you could also use PPI to prove Good’s theorem though. Presumably the reason it pays to get new evidence is that you should expect to assign more probability to the truth after observing new evidence?