I think the integral will converge when we use bounded scoring rules. The integral can be rewritten to have its integration limits finite with probability one, as it in practice integrates from 0 to the event time T, which should be assumed to be finite with probability 1. (The score is 0 from T to ∞, since p(t) equals the outcome at any time after T. I might not have been sufficiently clear about that though.)
Neat! I thought this was great!
As a nitpick for the integrated scoring rule, as written it looks like the integral doesn’t converge.
Some typos:
> Thus a rational forecaster will always expect the probability of a positive resolution to increase in time
should be decrease
> Example: “Will Putin be stay in power until August 11th 2030?”
be stay ⇒ stay
> Gompertz—Makeham
should have fewer --
Thank you!
I think the integral will converge when we use bounded scoring rules. The integral can be rewritten to have its integration limits finite with probability one, as it in practice integrates from 0 to the event time T, which should be assumed to be finite with probability 1. (The score is 0 from T to ∞, since p(t) equals the outcome at any time after T. I might not have been sufficiently clear about that though.)