Factorizable Joint Shift in Multinomial Classification

Factorizable joint shift was recently proposed as a type of dataset shift for which the characteristics can be estimated from observed data. For the multinomial (multi-class) classification setting, we derive a representation of factorizable joint shift in terms of the source (training) distribution, the target (test) prior class probabilities and the target marginal distribution of the features. On the basis of this result, we propose alternatives to joint importance aligning, at the same time pointing out the limitations encountered when making an assumption of factorizable joint shift. Other results of the paper include correction formulae for the posterior class probabilities both under general dataset shift and factorizable joint shift. In addition, we investigate the consequences of assuming factorizable joint shift for the bias caused by sample selection.