  We consider a a collection of categorical random variables. Of special
interest is the causal effect on an outcome variable following an intervention
on another variable. Conditionally on a Directed Acyclic Graph (DAG), we assume
that the joint law of the random variables can be factorized according to the
DAG, where each term is a categorical distribution for the node-variable given
a configuration of its parents. The graph is equipped with a causal
interpretation through the notion of interventional distribution and the allied
"do-calculus". From a modeling perspective, the likelihood is decomposed into a
product over nodes and parents of DAG-parameters, on which a suitably specified
collection of Dirichlet priors is assigned. The overall joint distribution on
the ensemble of DAG-parameters is then constructed using global and local
independence. We account for DAG-model uncertainty and propose a reversible
jump Markov Chain Monte Carlo (MCMC) algorithm which targets the joint
posterior over DAGs and DAG-parameters; from the output we are able to recover
a full posterior distribution of any causal effect coefficient of interest,
possibly summarized by a Bayesian Model Averaging (BMA) point estimate. We
validate our method through extensive simulation studies, wherein comparisons
with alternative state-of-the-art procedures reveal an outperformance in terms
of estimation accuracy. Finally, we analyze a dataset relative to a study on
depression and anxiety in undergraduate students.
