  We consider a binary response which is potentially affected by a set of
continuous variables. Of special interest is the causal effect on the response
due to an intervention on a specific variable. The latter can be meaningfully
determined on the basis of observational data through suitable assumptions on
the data generating mechanism. In particular we assume that the joint
distribution obeys the conditional independencies (Markov properties) inherent
in a Directed Acyclic Graph (DAG), and the DAG is given a causal interpretation
through the notion of interventional distribution. We propose a DAG-probit
model where the response is generated by discretization through a random
threshold of a continuous latent variable and the latter, jointly with the
remaining continuous variables, has a distribution belonging to a zero-mean
Gaussian model whose covariance matrix is constrained to satisfy the Markov
properties of the DAG. Our model leads to a natural definition of causal effect
conditionally on a given DAG. Since the DAG which generates the observations is
unknown, we present an efficient MCMC algorithm whose target is the posterior
distribution on the space of DAGs, the Cholesky parameters of the concentration
matrix, and the threshold linking the response to the latent. Our end result is
a Bayesian Model Averaging estimate of the causal effect which incorporates
parameter, as well as model, uncertainty. The methodology is assessed using
simulation experiments and applied to a gene expression data set originating
from breast cancer stem cells.
