Causal discovery from databases with discrete and continuous variables1

Causal discovery is widely used for analysis of experimental data focusing on the exploratory analysis and suggesting probable causal dependencies. There is a variety of causal discovery algorithms in the literature. Some of these algorithms rely on the assumption that there are no latent variables in the model; others do not provide a scoring metric to easily compare the reliability of two candidate models. Bayesian Constraint-based Causal Discovery (BCCD) [1] is a state-of-the-art-algorithm for causal discovery that tries to combine the strength of the best algorithms in the field. BCCD is able to detect latent variables in the model and determines the reliability of the edges between variables that makes it very easy to compare alternative models. The idea of BCCD is to estimate the reliability of causal relations by scoring Directed Acyclic Graphs (DAGs) for a smaller subset of variables using a Bayesian score and then to combine these statements to infer a final causal model. The Bayesian score has a closed form solution for discrete variables that makes the scoring of causal relations fast and efficient. The BCCD algorithm is currently limited to discrete or Gaussian variables as there is no closed form solution for the Bayesian score for a mixture of discrete and continuous variables. To extend BCCD, we need a new scoring method to estimate the reliability of causal relations. There are several scoring methods in the literature for mixtures of discrete and continuous variables. Most of these methods either rely on strict assumptions about the structure of the network that do not apply in practice, such as forbidding structures in the network with a continuous variable as a parent having a discrete variable as child, or are time consuming and/or memory inefficient.