  Several disciplines, like the social sciences, epidemiology, sentiment
analysis, or market research, are interested in knowing the distribution of the
classes in a population rather than the individual labels of the members
thereof. Quantification is the supervised machine learning task concerned with
obtaining accurate predictors of class prevalence, and to do so particularly in
the presence of label shift. The distribution-matching (DM) approaches
represent one of the most important families among the quantification methods
that have been proposed in the literature so far. Current DM approaches model
the involved populations by means of histograms of posterior probabilities. In
this paper, we argue that their application to the multiclass setting is
suboptimal since the histograms become class-specific, thus missing the
opportunity to model inter-class information that may exist in the data. We
propose a new representation mechanism based on multivariate densities that we
model via kernel density estimation (KDE). The experiments we have carried out
show our method, dubbed KDEy, yields superior quantification performance with
respect to previous DM approaches. We also investigate the KDE-based
representation within the maximum likelihood framework and show KDEy often
shows superior performance with respect to the expectation-maximization method
for quantification, arguably the strongest contender in the quantification
arena to date.
