We investigate the problem of classifying individuals based on estimated density functions for each individual. Given labelled histograms characterizing red blood cells (RBCs) for different individuals, the learning problem is to build a classifier which can classify new unlabelled histograms into normal and iron deficient classes. Thus, the problem is similar to conventional classification in that there is labelled training data, but different in that the underlying measurements are not feature vectors but histograms or density estimates. We describe a general framework based on probabilistic hierarchical models for modelling such data and illustrate how the model lends itself to classification. We contrast this approach with two other alternatives: (1) directly defining distance between densities using a cross-entropy distance measure, and (2) using parameters of the estimated densities as feature vectors for a standard discriminative classification framework. We evaluate all three methods on a real-world data set consisting of 180 subjects. The hierarchical modeling and density-distance approaches are most accurate, yielding cross-validated error rates in the range of 1 to 2%. We conclude by discussing the relative merits of each approach, including the interpretability of each model from a clinical diagnostic viewpoint.

Binning and truncation of data is common in data analysis and machine learning. This paper addresses the problem of fitting mixture densities to multivariate binned and truncated data. The EM approach proposed by McLachlan and Jones (1988) for the univariate case is generalized to multivariate measurements. The multivariate solution requires the evaluation of multidimensional integrals over each bin at each iteration of the EM procedure. Naive implementation of the procedure can lead to computationally inefficient results. To reduce the computational cost a number of straightforward numerical techniques are proposed. Results on simulated data indicate that the proposed methods can achieve significant computational gains with no loss in the accuracy of the final parameter estimates. Furthermore, experimental results suggest that with a sufficient number of bins and data points it is possible to estimate the true underlying density almost as well as if the data had not been binned. The paper concludes with a brief description of an application of this approach to diagnosis of iron deficiency anemia, in the context of binned and truncated bivariate measurements of volume and hemoglobin concentration from an individual's red blood cells.