  We consider the estimation of densities in multiple subpopulations, where the
available sample size in each subpopulation greatly varies. For example, in
epidemiology, different diseases may share similar pathogenic mechanism but
differ in their prevalence. Without specifying a parametric form, our proposed
approach pools information from the population and estimate the density in each
subpopulation in a data-driven fashion. Low-dimensional approximating density
families in the form of exponential families are constructed from the principal
modes of variation in the log-densities, within which subpopulation densities
are then fitted based on likelihood principles and shrinkage. The approximating
families increase in their flexibility as the number of components increases
and can approximate arbitrary infinite-dimensional densities with discrete
observations, for which we derived convergence results. The proposed methods
are shown to be interpretable and efficient in simulation as well as
applications to electronic medical record and rainfall data.
