Density Ratio Estimation and Neyman Pearson Classification with Missing Data

Density Ratio Estimation (DRE) is an important machine learning technique with many downstream applications. We consider the challenge of DRE with non-uniformly missing data. In this setting, we show that using standard DRE methods leads to biased results while our proposal (M-KLIEP), an adaptation of the popular DRE procedure KLIEP, restores consistency. Moreover, we provide finite sample estimation error bounds for M-KLIEP, which demonstrate minimax optimality with respect to both sample size and worst-case missingness. We then adapt an important downstream application of DRE, Neyman-Pearson (NP) classification, to this missing data setting. Our procedure both controls Type I error and achieves high power, with high probability. Finally, we demonstrate promising empirical performance on a range of both synthetic data and real world data with simulated missingness.