Context-Specific and Local Independence in Markovian Dependence Structures

Directed acyclic graphs (DAGs) have been established as one of the primary tools for characterizing dependencies and causality among variables in multivariate systems. However, it has also been recognized that DAGs may hide more nuanced forms of independence that are important for interpretation and operational efficiency of the dependence models. Such independencies are typically context-specific, meaning that a variable may lose its connection to another variable in a particular context determined by some other set of variables. Here we review context-specific independence in different classes of Markovian probability models both for static and spatially or temporally organized variables, including Bayesian networks, Markov networks, and higher-order Markov chains. The generality of the context-specific independence as a concept may spawn new ways to characterize dependence systems also beyond these traditional models, for example, in dependence logic.