  Causal discovery for purely observational, categorical data is a
long-standing challenging problem. Unlike continuous data, the vast majority of
existing methods for categorical data focus on inferring the Markov equivalence
class only, which leaves the direction of some causal relationships
undetermined. This paper proposes an identifiable ordinal causal discovery
method that exploits the ordinal information contained in many real-world
applications to uniquely identify the causal structure. The proposed method is
applicable beyond ordinal data via data discretization. Through real-world and
synthetic experiments, we demonstrate that the proposed ordinal causal
discovery method combined with simple score-and-search algorithms has favorable
and robust performance compared to state-of-the-art alternative methods in both
ordinal categorical and non-categorical data. An accompanied R package OrdCD is
freely available on CRAN and at
https://web.stat.tamu.edu/~yni/files/OrdCD_1.0.0.tar.gz.
