  Modern RNA sequencing technologies provide gene expression measurements from
single cells that promise refined insights on regulatory relationships among
genes. Directed graphical models are well-suited to explore such (cause-effect)
relationships. However, statistical analyses of single cell data are
complicated by the fact that the data often show zero-inflated expression
patterns. To address this challenge, we propose directed graphical models that
are based on Hurdle conditional distributions parametrized in terms of
polynomials in parent variables and their 0/1 indicators of being zero or
nonzero. While directed graphs for Gaussian models are only identifiable up to
an equivalence class in general, we show that, under a natural and weak
assumption, the exact directed acyclic graph of our zero-inflated models can be
identified. We propose methods for graph recovery, apply our model to real
single-cell RNA-seq data on T helper cells, and show simulated experiments that
validate the identifiability and graph estimation methods in practice.
