[Paper Review] Sequences of regressions and their independences

Nanny Wermuth, Kayvan Sadeghi|arXiv (Cornell University)|Mar 13, 2011

Bayesian Modeling and Causal Inference83 references4 citations

TL;DR

This paper introduces regression graphs as a generalization of directed acyclic graphs to model sequences of univariate or multivariate regressions, capturing conditional independences under Gaussian-like assumptions. It establishes polynomial-time algorithms for finding Markov equivalent DAGs, enabling efficient model selection and interpretation in causal inference and machine learning.

ABSTRACT

Ordered sequences of univariate or multivariate regressions provide statistical modelsfor analysingdata fromrandomized, possiblysequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, prove criteria for Markov equivalence anddiscussthenotion of simpler statistical covering models. Knowledgeof Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.

Motivation & Objective

• To formalize a graphical model framework—regression graphs—that extends directed acyclic graphs to represent sequences of regressions in observational and experimental studies.
• To characterize conditional independences in regression sequences under assumptions similar to multivariate Gaussian distributions.
• To develop criteria for Markov equivalence among regression graphs, enabling alternative interpretations of the same statistical model.
• To provide a polynomial-time algorithm for identifying a Markov equivalent directed acyclic graph from a given regression graph.
• To support causal discovery and statistical modeling by simplifying complex graphical criteria through equivalence classes.

Proposed method

• Introduces two types of undirected edges in regression graphs: one for components of the joint response variable and another for components of the context vector.
• Defines the Markov property for regression graphs, linking d-separation in the graph to conditional independence in the probability distribution.
• Establishes criteria for Markov equivalence between regression graphs using the structure of v-structures and unshielded colliders.
• Proves that every regression graph has at least one Markov equivalent directed acyclic graph under standard assumptions.
• Develops a polynomial-time algorithm to compute a Markov equivalent DAG from a given regression graph, leveraging equivalence class structure.
• Uses graphical criteria to simplify model selection and inference, avoiding complex conditional independence testing.

Experimental results

Research questions

• RQ1 How can sequences of regressions be represented using a unified graphical model that captures conditional independences?
• RQ2 What are the conditions under which a regression graph is Markov equivalent to a directed acyclic graph?
• RQ3 Can a polynomial-time algorithm be constructed to find a Markov equivalent DAG for any given regression graph?
• RQ4 How do the structural features of regression graphs—particularly the two types of undirected edges—relate to statistical independences?
• RQ5 In what ways does Markov equivalence in regression graphs improve model interpretation and machine learning applications?

Key findings

• Regression graphs extend directed acyclic graphs by incorporating two types of undirected edges, enabling representation of joint responses and context variables.
• Conditional independences in regression sequences are d-separation equivalent to those in the corresponding regression graph under Gaussian-like assumptions.
• Markov equivalence between regression graphs can be characterized using structural features such as v-structures and unshielded colliders.
• A polynomial-time algorithm exists to compute a Markov equivalent directed acyclic graph for any given regression graph.
• The existence of Markov equivalent DAGs allows for alternative interpretations of the same regression sequence, enhancing model flexibility.
• The graphical criteria for regression graphs simplify the identification of conditional independence structures compared to general graphical models.