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Reductions of discrete Bayesian networks via lumping

Linard Hoessly

TL;DR

This work uses surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure, discussing the connection between this and reductions of homogeneous and non-homogeneous Markov chains.

Abstract

Bayesian networks are widely utilised in various fields, offering elegant representations of factorisations and causal relationships. We use surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure. We introduce and define corresponding notions, analyse their properties, and provide examples of highly symmetric special cases, enhancing the understanding of the fundamental properties of such reductions for Bayesian networks. We also discuss the connection between this and reductions of homogeneous and non-homogeneous Markov chains.

Reductions of discrete Bayesian networks via lumping

TL;DR

This work uses surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure, discussing the connection between this and reductions of homogeneous and non-homogeneous Markov chains.

Abstract

Bayesian networks are widely utilised in various fields, offering elegant representations of factorisations and causal relationships. We use surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure. We introduce and define corresponding notions, analyse their properties, and provide examples of highly symmetric special cases, enhancing the understanding of the fundamental properties of such reductions for Bayesian networks. We also discuss the connection between this and reductions of homogeneous and non-homogeneous Markov chains.
Paper Structure (26 sections, 25 theorems, 89 equations)

This paper contains 26 sections, 25 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Previous approaches
  3. Content
  4. Preliminaries
  5. Graph theory and notation for DAGs
  6. Notations
  7. Bayesian networks
  8. Lumping of directed graphical models
  9. DTMCs as BNs
  10. Results
  11. Generalities on projected BNs
  12. General observations
  13. Results for \ref{['D1']}
  14. Results for \ref{['D2']}
  15. Results for \ref{['D3']}
  16. ...and 11 more sections

Key Result

Theorem 1

lauritzen1996 Let $(X_v)_{v\in V}$ be a discrete random vector. The following are equivalent: - $(X_v)_{v\in V}$ factorises over the DAG $\mathcal{G}$. - (Local Markov property) For any $v\in V$, $(X_v)_{v\in V}$ satisfies - (Global Markov property) For any triple $(A,B,S)$ of disjoint subsets of $V$ such that $S$ d-separates $A$ from $B$ in $\mathcal{G}$, $(X_v)_{v\in V}$ satisfies

Theorems & Definitions (53)

  • Theorem 1
  • Remark 1
  • Definition 2
  • Example 1
  • Remark 2
  • Example 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5
  • ...and 43 more