Minimal Assumptions for Optimal Serology Classification: Theory and
  Implications for Multidimensional Settings and Impure Training Data
Paul N. Patrone Raquel A. Binder Catherine S. Forconi Ann M. Moormann Anthony J. Kearsley
Minimizing error in prevalence estimates and diagnostic classifiers remains a
challenging task in serology. In theory, these problems can be reduced to
modeling class-conditional probability densities (PDFs) of measurement
outcomes, which control all downstream analyses. However, this task quickly
succumbs to the curse of dimensionality, even for assay outputs with only a few
dimensions (e.g. target antigens). To address this problem, we propose a
technique that uses empirical training data to classify samples and estimate
prevalence in arbitrary dimension without direct access to the conditional
PDFs. We motivate this method via a lemma that relates relative conditional
probabilities to minimum-error classification boundaries. This leads us to
formulate an optimization problem that: (i) embeds the data in a parameterized,
curved space; (ii) classifies samples based on their position relative to a
coordinate axis; and (iii) subsequently optimizes the space by minimizing the
empirical classification error of pure training data, for which the classes are
known. Interestingly, the solution to this problem requires use of a
homotopy-type method to stabilize the optimization. We then extend the analysis
to the case of impure training data, for which the classes are unknown. We find
that two impure datasets suffice for both prevalence estimation and
classification, provided they satisfy a linear independence property. Lastly,
we discuss how our analysis unifies discriminative and generative learning
techniques in a common framework based on ideas from set and measure theory.
Throughout, we validate our methods in the context of synthetic data and a
research-use SARS-CoV-2 enzyme-linked immunosorbent (ELISA) assay.