That’s sufficient information to calculate the conditional prediction curves I’m proposing. What you need is Ps(X=1∣T≥t). If you have Ps(X=1) and Ps(T≥t), which you can find by integrating the density for “when will X happen”, you can calculate Ps(X=1∣T≥t).
Perhaps the way to enable forecasters to have automatically-updating forecasts based on the passage of time is to ask questions in pairs:
(1) Will X happen by Date?
(2) Conditional on X happening by Date, when will X happen?
The probability density function a forecaster gives for 2 can then be used to auto-update their binary forecast for 1.
I only skimmed the post and didn’t try to follow the math, so I’m not sure if you already made this point in the post.
That’s sufficient information to calculate the conditional prediction curves I’m proposing. What you need is Ps(X=1∣T≥t). If you have Ps(X=1) and Ps(T≥t), which you can find by integrating the density for “when will X happen”, you can calculate Ps(X=1∣T≥t).