Markov Properties for Graphical Models with Cycles and Latent Variables

Abstract

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.

arXiv comment131 pages

References in corpus (7)

• Beyond Bell's Theorem: Correlation Scenarios
• Graphs for margins of Bayesian networks
• Markov properties for mixed graphs
• Markovian acyclic directed mixed graphs for discrete data
• Margins of discrete Bayesian networks
• On Deducing Conditional Independence from d-Separation in Causal Graphs with Feedback (Research Note)
• Marginalization and Conditioning for LWF Chain Graphs

Cited by in corpus (14)

• Foundations of Structural Causal Models with Cycles and Latent Variables
• Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounders
• A general framework for cyclic and fine-tuned causal models and their compatibility with space-time
• Impossibility of superluminal signalling in Minkowski space-time does not rule out causal loops
• Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias
• Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles
• Boosting Local Causal Discovery in High-Dimensional Expression Data
• Robustness of Model Predictions under Extension
• The Global Markov Property for a Mixture of DAGs
• A Critique on the Interventional Detection of Causal Relationships
• Universal Decision Models
• Conditions and Assumptions for Constraint-based Causal Structure Learning
• The standard no-signalling constraints in Bell scenarios are neither sufficient nor necessary for preventing superluminal signalling with general interventions
• On the Foundations of Cycles in Bayesian Networks