Labeled Directed Acyclic Graphs: a generalization of context-specific independence in directed graphical models

Directed acyclic graphs have gained widespread popularity as representations of complex multivariate systems (Koski and Noble (2009); Koller and Friedman (2009)). Despite their advantageous properties for representing dependencies among variables in a modular fashion, several proposals for making them more flexible and parsimonious have been presented (Boutilier et al (1996); Friedman and Goldszmidt (1996); Chickering et al (1997); Eriksen (1999); Poole and Zhang (2003); Koller and Friedman (2009)). In particular, an important notion is to allow the dependencies to have local structures, such that a node need not explicitly depend on all the combinations of values of its parents. This leads to contextspecific independence which can substantially reduce the parametric dimensionality of a network model and lead to a more expressive interpretation of the dependence structure (Boutilier et al (1996); Friedman and Goldszmidt (1996); Poole and Zhang (2003); Koller and Friedman (2009)). Contextspecific independencies have also been seemingly separately considered for undirected graphical models by multiple authors (Corander (2003); Højsgaard (2003, 2004)).