The relation between Granger causality and directed information theory: a review <s> D. More on graphs <s> We introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series. The modelling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear dependences. The models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs. In these graphs each component series is represented by a single vertex and directed edges indicate possible Granger-causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure. We introduce various notions of Granger-causal Markov properties and discuss the relationships among them and to other Markov properties that can be applied in this context. Examples for graphical time series models include nonlinear autoregressive models and multivariate ARCH models. <s> BIB001 </s> The relation between Granger causality and directed information theory: a review <s> D. More on graphs <s> We propose a new type of probabilistic graphical model, based on directed information, to represent the causal dynamics between processes in a stochastic system. We show the practical significance of such graphs by proving their equivalence to generative model graphs which succinctly summarize interdependencies for causal dynamical systems under mild assumptions. This equivalence means that directed information graphs may be used for causal inference and learning tasks in the same manner Bayesian networks are used for correlative statistical inference and learning. <s> BIB002
Granger causality graphs were defined and studied in BIB001 . A causality graph is a mixed graph (V, E d , E u ) that encodes Granger causality relationships between the components of x V . The vertex set V stores the indexes of the components of x V . E d is a set of directed edges beween vertices. A directed edge from a to b is equivalent to "x a Granger causes x b relatively to V ". E u is a set of undirected edges. An undirected edge between x a and x b is equivalent to "x a and x b are (conditionally if def.4 adopted) instantaneously coupled". Interestingly, a DRAFT Granger causality graph may have Markov properties (as in usual graphical models) reflecting a particular (spatial) structure of the joint probability distribution of the whole process {x t V } BIB001 . A taxonomy of Markov properties: local, global, block recursive is studied in BIB001 , and equivalence between these properties is put forward. More interestingly, these properties are linked with topological properties of the graph. Therefore, structural properties of the graphs are equivalent to a particular factorization of the joint probability of the multivariate process. We will not continue on this subject here, but this must be known since it paves the way to more efficient inference methods for Granger graphical modeling of multivariate processes (see a first step in this direction in BIB002 ).