Linking losses for density ratio and class-probability estimation

Supplementary material for “Linking losses for density ratio and class-probability estimation” A. The link Ψdr when π 6= 12 Our discussion in §4.1 assumed that π = 12 , as the KLIEP and LSIF risks (Equations 11, 13) would otherwise require scaling each of the expectations. But in general, we expect π 6= 12 . Of course, since the role of π is just to scale the link Ψdr by a constant, one may legitimately ignore its impact on the Bayes-optimal scorer. But for completeness, the general case may be analysed as follows. A common way of simulating balanced classes is by weighting the loss by the inverse of the class proportions, i.e. constructing `bal,−1(v) = (1− π)−1 · `−1(v) and `bal,1(v) = π−1 · `1(v).