Sequences of regressions and their independences

TL;DR

This paper explores regression graphs that model conditional independences in complex data, providing criteria for reading independences, establishing Markov equivalence, and offering algorithms for graph simplification.

Contribution

It introduces new criteria for reading independences from regression graphs, characterizes Markov equivalence, and presents a polynomial time algorithm for finding equivalent directed acyclic graphs.

Findings

Derived criteria to interpret all implied independences.

Proved criteria for Markov equivalence between graphs.

Developed a polynomial time algorithm for finding Markov equivalent DAGs.

Abstract

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential 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, derive criteria to read all implied independences of a regression graph and prove criteria for Markov equivalence that is to judge whether two different graphs imply the same set of independence statements. Knowledge of 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.