Maybe I'm missing something, but if we are estimating the P(Xi), how can we also have Xi on RHS? and what is the adjustment +(1−Xi)(1−qi,j). why is that there?
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.
Maybe I'm missing something, but if we are estimating the P(Xi), how can we also have Xi on RHS? and what is the adjustment +(1−Xi)(1−qi,j). why is that there?