Approximate Implication for Probabilistic Graphical Models

The graphical structure of Probabilistic Graphical Models (PGMs) represents the conditional independence (CI) relations that hold in the modeled distribution. Every separator in the graph represents a conditional independence relation in the distribution, making them the vehicle through which new conditional independencies are inferred and verified. The notion of separation in graphs depends on whether the graph is directed (i.e., a Bayesian Network), or undirected (i.e., a Markov Network). The premise of all current systems-of-inference for deriving CIs in PGMs, is that the set of CIs used for the construction of the PGM hold exactly. In practice, algorithms for extracting the structure of PGMs from data discover approximate CIs that do not hold exactly in the distribution. In this paper, we ask how the error in this set propagates to the inferred CIs read off the graphical structure. More precisely, what guarantee can we provide on the inferred CI when the set of CIs that entailed it hold only approximately? It has recently been shown that in the general case, no such guarantee can be provided. In this work, we prove new negative and positive results concerning this problem. We prove that separators in undirected PGMs do not necessarily represent approximate CIs. That is, no guarantee can be provided for CIs inferred from the structure of undirected graphs. We prove that such a guarantee exists for the set of CIs inferred in directed graphical models, making the $d$-separation algorithm a sound and complete system for inferring approximate CIs. We also establish improved approximation guarantees for independence relations derived from marginal and saturated CIs.