Probabilistic exponential family inverse regression and its applications

Rapid advances in high-throughput sequencing technologies have led to the fast accumulation of high-dimensional data, which is harnessed for understanding the implications of various factors on human disease and health. While dimension reduction plays an essential role in high-dimensional regression and classification, existing methods often require the predictors to be continuous, making them unsuitable for discrete data, such as presence-absence records of species in community ecology and sequencing reads in single-cell studies. To identify and estimate sufficient reductions in regressions with discrete predictors, we introduce probabilistic exponential family inverse regression (PrEFIR), assuming that, given the response and a set of latent factors, the predictors follow one-parameter exponential families. We show that the low-dimensional reductions result not only from the response variable but also from the latent factors. We further extend the latent factor modeling framework to the double exponential family by including an additional parameter to account for the dispersion. This versatile framework encompasses regressions with all categorical or a mixture of categorical and continuous predictors. We propose the method of maximum hierarchical likelihood for estimation, and develop a highly parallelizable algorithm for its computation. The effectiveness of PrEFIR is demonstrated through simulation studies and real data examples.