February 22, 2019
14:30 - 15:30
ISBA - C115 (Seminar Room Bernoulli)
Eugen Pircalabelu, ISBA - UCLouvain
"Using vine copulas to estimate the structure of directed acyclical graphs"
We present a new method of estimating and selecting a Bayesian network for continuous data with the goal of stepping outside the class of multivariate normal distributions which are generally used due to their attractive properties. Bayesian networks comprise both a graphical representation in the form of a directed acyclic graph (DAG) for which all the edges between the nodes have a single direction and no loops are allowed, and the decomposition of the joint probability density function as a product of conditional and marginal density functions according to the graph and this property makes such a class of statistical models useful for simplifying general multivariate densities, because the absence of edges between two nodes can be interpreted as a conditional independence between the variables associated with those nodes conditional on other random variables.
A different technique for analyzing multivariate dependencies is by using copulas functions. The key aspect distinguishing copulas from Bayesian networks is that, when modeling data using copulas, one is generally focusing on the probabilistic aspect without special attention to the additional information provided by the graph or directionality of edges between nodes.
Our proposed procedure combines directed acyclic graphs and their associated probability models with copula C/D vines in order to construct `copula based DAGs' which allow more flexibility in modeling joint distributions of pairs of nodes in the network. We exploit connections and similarities that exist between these two statistical techniques with the explicit purpose of estimating a directed graphical model, a network, for continuous data that are not necessarily normally distributed. The approach uses a score based learning scheme, where one modifies an initial graph based on improvements in the score, until a local maximum score is reached.
A new information criterion is proposed and studied for graph selection tailored to the joint modeling of data based on graphs and copulas. Examples and simulation studies show the flexibility and properties of the method.