  Estimating causal models from observational data is a crucial task in data
analysis. For continuous-valued data, Shimizu et al. have proposed a linear
acyclic non-Gaussian model to understand the data generating process, and have
shown that their model is identifiable when the number of data is sufficiently
large. However, situations in which continuous and discrete variables coexist
in the same problem are common in practice. Most existing causal discovery
methods either ignore the discrete data and apply a continuous-valued algorithm
or discretize all the continuous data and then apply a discrete Bayesian
network approach. These methods possibly loss important information when we
ignore discrete data or introduce the approximation error due to
discretization. In this paper, we define a novel hybrid causal model which
consists of both continuous and discrete variables. The model assumes: (1) the
value of a continuous variable is a linear function of its parent variables
plus a non-Gaussian noise, and (2) each discrete variable is a logistic
variable whose distribution parameters depend on the values of its parent
variables. In addition, we derive the BIC scoring function for model selection.
The new discovery algorithm can learn causal structures from mixed continuous
and discrete data without discretization. We empirically demonstrate the power
of our method through thorough simulations.
