Approximate Implication with d-Separation

TL;DR

This paper investigates how errors in the estimated conditional independencies in probabilistic graphical models affect the accuracy of inferred independencies, providing guarantees for directed models.

Contribution

It proves that d-separation can be sound and complete for approximate CIs in directed graphical models, unlike the general case.

Findings

Guarantee exists for approximate CIs in directed models

d-separation is sound and complete for approximate inference

Provides approximation guarantees for marginal CIs

Abstract

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.