Log-likelihood invariance in markov equivalent bayesian networks

Abstract

Bayesian Networks are widely used to model probabilistic relationships between variables through DAGs. However, multiple Bayesian Networks can represent the same set of conditional independencies, forming what is known as a MEC. This presents a challenge in model evaluation, as networks within the same MEC are statistically indistinguishable based on observational data alone. This work investigates whether the log-likelihood function, a standard metric for assessing model fit, remains invariant across all Bayesian Networks in a given MEC. After learning the CPDAG of a Bayesian Network from data, a method is implemented to enumerate DAGs in the corresponding MEC, converting each DAG to a Bayesian Network and compute the log-likelihood for each structure using Bayesian parameter estimation. Structural modifications are also introduced to produce networks outside the original MEC, enabling comparison of log-likelihood values across different equivalence classes. Experiments conducted on benchmark networks (Asia, ALARM, Hepar2) confirm that log-likelihood values remain constant within each MEC and vary significantly when the structure is modified to break v-structures. These findings suggest that log-likelihood invariance can serve as a reliable indicator of MEC membership for Bayesian Networks, allowing for more efficient model evaluation and structure learning without full enumeration.