A Transformational Characterization of Unconditionally Equivalent Bayesian Networks

TL;DR

This paper provides a transformational characterization of Bayesian networks up to unconditional equivalence, enabling easier identification and analysis of their equivalence classes through graph transformations.

Contribution

It introduces a novel characterization of unconditional equivalence classes of Bayesian networks using graph transformations and extends this to essential graphs representing Markov equivalence classes.

Findings

Unconditional equivalence classes are represented by undirected graphs with clique structures.

Two DAGs are in the same UEC if one can be transformed into the other via specific moves.

The characterization aids in estimating equivalence classes from marginal independence tests.

Abstract

We consider the problem of characterizing Bayesian networks up to unconditional equivalence, i.e., when directed acyclic graphs (DAGs) have the same set of unconditional $d$-separation statements. Each unconditional equivalence class (UEC) is uniquely represented with an undirected graph whose clique structure encodes the members of the class. Via this structure, we provide a transformational characterization of unconditional equivalence; i.e., we show that two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of specified moves. We also extend this characterization to the essential graphs representing the Markov equivalence classes (MECs) in the UEC. UECs partition the space of MECs and are easily estimable from marginal independence tests. Thus, a characterization of unconditional equivalence has applications in methods that involve searching the space of MECs of Bayesian networks.