The purpose of class distribution estimation (also known as quantification) is to determine the values of the prior class probabilities in a test dataset without class label observations. A variety of methods to achieve this have been proposed in the literature, most of them based on the assumption that the distributions of the training and test data are related through prior probability shift (also known as label shift). Among these methods, Friedman's method has recently been found to perform relatively well both for binary and multi-class quantification. We discuss the properties of Friedman's method and another approach mentioned by Friedman (called DeBias method in the literature) in the context of a general framework for designing linear equation systems for class distribution estimation.

We study the problem of class distribution estimation under dataset shift. On the training dataset, both features and class labels are observed while on the test dataset only the features can be observed. The task then is the estimation of the distribution of the class labels, i.e. the estimation of the class prior probabilities, in the test dataset. Assumptions of invariance between the training joint distribution of features and labels and the test distribution can considerably facilitate this task. We discuss the assumptions of covariate shift, factorizable joint shift, and sparse joint shift and their implications for class distribution estimation.