Approximate Implication with d-Separation
This addresses a foundational issue in machine learning for researchers using PGMs, providing theoretical guarantees for approximate inference, though it is incremental as it builds on prior impossibility results.
The paper tackles the problem of how errors in approximate conditional independencies (CIs) used to construct probabilistic graphical models propagate to inferred CIs via d-separation, proving that a guarantee exists for directed graphical models, making d-separation sound and complete for approximate CIs.
The graphical structure of Probabilistic Graphical Models (PGMs) encodes the conditional independence (CI) relations that hold in the modeled distribution. Graph algorithms, such as d-separation, use this structure to infer additional conditional independencies, and to query whether a specific CI holds in the distribution. 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. 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 prove an approximation guarantee for independence relations derived from marginal CIs.