Markov Properties for Graphical Models with Cycles and Latent Variables
This work addresses foundational issues in graphical modeling for researchers in statistics and machine learning, though it appears incremental as it extends prior frameworks without demonstrating new applications or empirical results.
The paper tackles the problem of defining Markov properties for probabilistic graphical models that include cycles and latent variables, introducing HEDGes as a generalization of existing models and showing that these properties are not equivalent in such complex settings.
We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that can model cycles. We define and analyse several different Markov properties that relate the graphical structure of a HEDG with a probability distribution on a corresponding product space over the set of nodes, for example factorization properties, structural equations properties, ordered/local/global Markov properties, and marginal versions of these. The various Markov properties for HEDGes are in general not equivalent to each other when cycles or hyperedges are present, in contrast with the simpler case of directed acyclic graphical (DAG) models (also known as Bayesian networks). We show how the Markov properties for HEDGes - and thus the corresponding graphical Markov models - are logically related to each other.