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Approximate Implication for Probabilistic Graphical Models

Batya Kenig

TL;DR

The paper tackles how approximate conditional independencies (CIs) propagate through probabilistic graphical models (PGMs) when CI statements inferred from data do not hold exactly. Using an information-theoretic framework with entropy $H$ and conditional mutual information $I$, it derives both negative and positive results: for undirected Markov networks, separators do not guarantee AI; for directed Bayesian networks, the $d$-separation criterion and the recursive basis yield a sound 1-approximation for inferred CIs, with tighter bounds for saturated and marginal CIs. It further strengthens the theory via the I-measure, establishing when EI implies AI within positive polymatroids and general polymatroids, and proves that the intersection axiom does not relax AI in general. The work provides concrete, computable relaxation factors (e.g., $h( au) \nleq rac{n}{2} h(igcup ext{antecedents})$ for saturated CIs and a 1-approximation for recursive CIs), with tightness results and practical implications for structure learning under approximate CI assumptions. Overall, it clarifies when reliable approximate CI inference is possible and quantifies the bounds under different derivation rules and CI families, guiding robust PGM structure learning from data.

Abstract

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.

Approximate Implication for Probabilistic Graphical Models

TL;DR

The paper tackles how approximate conditional independencies (CIs) propagate through probabilistic graphical models (PGMs) when CI statements inferred from data do not hold exactly. Using an information-theoretic framework with entropy and conditional mutual information , it derives both negative and positive results: for undirected Markov networks, separators do not guarantee AI; for directed Bayesian networks, the -separation criterion and the recursive basis yield a sound 1-approximation for inferred CIs, with tighter bounds for saturated and marginal CIs. It further strengthens the theory via the I-measure, establishing when EI implies AI within positive polymatroids and general polymatroids, and proves that the intersection axiom does not relax AI in general. The work provides concrete, computable relaxation factors (e.g., for saturated CIs and a 1-approximation for recursive CIs), with tightness results and practical implications for structure learning under approximate CI assumptions. Overall, it clarifies when reliable approximate CI inference is possible and quantifies the bounds under different derivation rules and CI families, guiding robust PGM structure learning from data.

Abstract

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 -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.
Paper Structure (22 sections, 19 theorems, 82 equations, 1 figure, 3 tables)

This paper contains 22 sections, 19 theorems, 82 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conditional Independence
  4. Background on Information Theory
  5. Exact Implication for Probabilistic Graphical Models
  6. Markov Networks and Saturated Independence
  7. Bayesian Networks
  8. Intersection Axiom Does not Relax
  9. Properties of Exact Implication
  10. The I-measure
  11. Exact implication in the set of positive polymatroids
  12. Exact Implication in the set of polymatroids
  13. Approximate Implication For Saturated CIs
  14. Proof of Lemma \ref{['lem:neqSaturated']}
  15. Approximate Implication for Recursive CIs
  16. ...and 7 more sections

Key Result

Theorem 3.4

GeigerPearl1993DBLP:conf/sigmod/BeeriFH77 Let $\Sigma$ be a set of saturated CIs over the set $\Omega\mathrel{\stackrel{\textsf{\tiny def}}{=}} \mathord{\{X_1,\dots,X_n\}}$ of variables, and let $\Sigma^+$ denote the closure of $\Sigma$ with respect to the semi-graphoid axioms. Let $\tau$ be a satur

Figures (1)

  • Figure 1: The information diagram for the joint probability $p(A,B,C)$ where $A,B$ and $C$ are defined in \ref{['eq:A']}-\ref{['eq:C']}.

Theorems & Definitions (42)

  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 32 more