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Conditional Independence under Infinite Measures and Poisson Point Processes

Shuyang Bai, Vishal Routh

Abstract

We study conditional independence under infinite measures on punctured product spaces, a notion recently introduced for graphical modeling in multivariate extremes and Lévy processes. In contrast to classical probabilistic conditional independence, this concept is formulated through normalized restrictions of an infinite measure that reflects the non-product structure of the punctured space. We show that this non-standard notion admits a natural probabilistic characterization: it is equivalent to classical conditional independence between coordinate projections of a Poisson point process defined on the punctured space with the given infinite measure as its mean measure. In addition, we provide a functional characterization of the conditional independence concept at the level of the enumerated points of the Poisson point process. We further extend the framework from punctured Euclidean product spaces to a more general abstract setting, thereby broadening its scope of potential applications.

Conditional Independence under Infinite Measures and Poisson Point Processes

Abstract

We study conditional independence under infinite measures on punctured product spaces, a notion recently introduced for graphical modeling in multivariate extremes and Lévy processes. In contrast to classical probabilistic conditional independence, this concept is formulated through normalized restrictions of an infinite measure that reflects the non-product structure of the punctured space. We show that this non-standard notion admits a natural probabilistic characterization: it is equivalent to classical conditional independence between coordinate projections of a Poisson point process defined on the punctured space with the given infinite measure as its mean measure. In addition, we provide a functional characterization of the conditional independence concept at the level of the enumerated points of the Poisson point process. We further extend the framework from punctured Euclidean product spaces to a more general abstract setting, thereby broadening its scope of potential applications.

Paper Structure

This paper contains 6 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction and overview of results
  2. Preliminaries
  3. Conditional independence of projections of Poisson point processes
  4. Non-punctured product space
  5. Punctured product spaces
  6. General dimensions

Key Result

Theorem 1.2

Suppose the assumption eq:E1 holds, and we have disjoint subsets $A, B, C \subseteq V$. Then the conditional independence relation in Definition Def:Lambda cond indep holds if and only if Here, when $C = \emptyset$, the relation is understood as unconditional independence, and when $A=\emptyset$ or $B=\emptyset$, the conditional or unconditional independence relation is understood to hold trivia

Theorems & Definitions (28)

  • Definition 1.1: engelke2025graphical, Definition 3.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1: Uniqueness
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 18 more