  We report a scalable hybrid quantum-classical machine learning framework to
build Bayesian networks (BN) that captures the conditional dependence and
causal relationships of random variables. The generation of a BN consists of
finding a directed acyclic graph (DAG) and the associated joint probability
distribution of the nodes consistent with a given dataset. This is a
combinatorial problem of structural learning of the underlying graph, starting
from a single node and building one arc at a time, that fits a given ensemble
using maximum likelihood estimators (MLE). It is cast as an optimization
problem that consists of a scoring step performed on a classical computer,
penalties for acyclicity and number of parents allowed constraints, and a
search step implemented using a quantum annealer. We have assumed uniform
priors in deriving the Bayesian network that can be relaxed by formulating the
problem as an estimation Dirichlet parameters. We demonstrate the utility of
the framework by applying to the problem of elucidating the gene regulatory
network for the MAPK/Raf pathway in human T-cells using proteomics data where
the concentration of proteins, nodes of the BN, are interpreted as
probabilities.
