Maybe I’m missing something, but if we are estimating the P(Xi), how can we also have Xi on RHS?
These probabilities are used for scoring predictions over the observed variables once the market resolves, so at that point we “don’t need” P(Xi) because we already know what Xi is. The only reason we compute it is so we can reward people who got the prediction right long ago before Xi was known.
and what is the adjustment +(1−Xi)(1−qi,j). why is that there?
Xiqi,j+(1−Xi)(1−qi,j) is equivalent to “qi,j if Xi = 1; otherwise 1-qi,j if Xi = 0”. It’s basically a way to mathematize the “contigency table” aspect.
These probabilities are used for scoring predictions over the observed variables once the market resolves, so at that point we “don’t need” P(Xi) because we already know what Xi is. The only reason we compute it is so we can reward people who got the prediction right long ago before Xi was known.
Xiqi,j+(1−Xi)(1−qi,j) is equivalent to “qi,j if Xi = 1; otherwise 1-qi,j if Xi = 0”. It’s basically a way to mathematize the “contigency table” aspect.