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Learning Bayesian and Markov Networks with an Unreliable Oracle

Juha Harviainen, Pekka Parviainen, Vidya Sagar Sharma

TL;DR

This work studies constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle and proves that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded.

Abstract

We study constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle that makes at most a bounded number of errors. For Markov networks, we observe that a low maximum number of vertex-wise disjoint paths implies that the structure is uniquely identifiable even if the number of errors is (moderately) exponential in the number of vertices. For Bayesian networks, however, we prove that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded. Finally, we give algorithms for structure learning when the structure is uniquely identifiable.

Learning Bayesian and Markov Networks with an Unreliable Oracle

TL;DR

This work studies constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle and proves that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded.

Abstract

We study constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle that makes at most a bounded number of errors. For Markov networks, we observe that a low maximum number of vertex-wise disjoint paths implies that the structure is uniquely identifiable even if the number of errors is (moderately) exponential in the number of vertices. For Bayesian networks, however, we prove that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded. Finally, we give algorithms for structure learning when the structure is uniquely identifiable.
Paper Structure (10 sections, 10 theorems, 5 equations, 4 figures, 1 table)

This paper contains 10 sections, 10 theorems, 5 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $k$-Identifiability
  4. Parameterised bounds
  5. Markov networks
  6. Bayesian networks
  7. Finding nearest neighbors
  8. Learning with an unreliable oracle
  9. Concluding Remarks
  10. Complete Proofs

Key Result

Lemma 1

Let $\kappa(G)$ be the maximum pairwise connectivity of $G$. Given two distinct undirected graphs $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$, there are at least $2^{n - 2 - \kappa(G_1)}$ separation statements with different outputs for $G_1$ and $G_2$.

Figures (4)

  • Figure 1: Two sparse graphs whose d-separation disctance is 1.
  • Figure 2: Skeletons of $D$ and $D'$.
  • Figure 3: Examples of $D$ and $D'$.
  • Figure 4: Possible skeletons of $D"$.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Definition 3.1: Table Function for a Bayesian Network
  • Definition 3.2: distance between two Bayesian Networks
  • Definition 3.3: Closest Bayesian Network
  • Lemma 2
  • proof
  • ...and 11 more