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Tree-structured Markov random fields with Poisson marginal distributions

Benjamin Côté, Hélène Cossette, Etienne Marceau

TL;DR

This work addresses modeling counts with tractable dependence by introducing a tree-structured multivariate Poisson MRF that preserves Poisson marginals with parameter $\lambda$ while encoding dependence along edges through $\alpha_e$. The authors provide a stochastic construction based on binomial thinning that yields explicit joint PMF and PGF, enabling exact sampling and efficient computation of the sum distribution, which is shown to be compound Poisson with a tree-structured secondary distribution. They establish positive dependence along edges, derive covariance along paths, and prove ordering results (supermodular and convex) that describe how increasing dependence affects the aggregate $M=\sum_v N_v$ under fixed trees, with no general order across different topologies. The framework supports fast FFT-based computations of the distribution of $M$ and the ordinary generating function of expected allocations (OGFEA), facilitating risk-sharing and allocation analyses for high-dimensional count data. The work also highlights the impact of tree topology on dependence and suggests directions for extending the approach to general graphs and topology-aware analyses of centrality and ordering.

Abstract

A new family of tree-structured Markov random fields for a vector of discrete counting random variables is introduced. According to the characteristics of the family, the marginal distributions of the Markov random fields are all Poisson with the same mean, and are untied from the strength or structure of their built-in dependence. This key feature is uncommon for Markov random fields and most convenient for applications purposes. The specific properties of this new family confer a straightforward sampling procedure and analytic expressions for the joint probability mass function and the joint probability generating function of the vector of counting random variables, thus granting computational methods that scale well to vectors of high dimension. We study the distribution of the sum of random variables constituting a Markov random field from the proposed family, analyze a random variable's individual contribution to that sum through expected allocations, and establish stochastic orderings to assess a wide understanding of their behavior.

Tree-structured Markov random fields with Poisson marginal distributions

TL;DR

This work addresses modeling counts with tractable dependence by introducing a tree-structured multivariate Poisson MRF that preserves Poisson marginals with parameter while encoding dependence along edges through . The authors provide a stochastic construction based on binomial thinning that yields explicit joint PMF and PGF, enabling exact sampling and efficient computation of the sum distribution, which is shown to be compound Poisson with a tree-structured secondary distribution. They establish positive dependence along edges, derive covariance along paths, and prove ordering results (supermodular and convex) that describe how increasing dependence affects the aggregate under fixed trees, with no general order across different topologies. The framework supports fast FFT-based computations of the distribution of and the ordinary generating function of expected allocations (OGFEA), facilitating risk-sharing and allocation analyses for high-dimensional count data. The work also highlights the impact of tree topology on dependence and suggests directions for extending the approach to general graphs and topology-aware analyses of centrality and ordering.

Abstract

A new family of tree-structured Markov random fields for a vector of discrete counting random variables is introduced. According to the characteristics of the family, the marginal distributions of the Markov random fields are all Poisson with the same mean, and are untied from the strength or structure of their built-in dependence. This key feature is uncommon for Markov random fields and most convenient for applications purposes. The specific properties of this new family confer a straightforward sampling procedure and analytic expressions for the joint probability mass function and the joint probability generating function of the vector of counting random variables, thus granting computational methods that scale well to vectors of high dimension. We study the distribution of the sum of random variables constituting a Markov random field from the proposed family, analyze a random variable's individual contribution to that sum through expected allocations, and establish stochastic orderings to assess a wide understanding of their behavior.
Paper Structure (16 sections, 13 theorems, 75 equations, 10 figures, 4 tables, 5 algorithms)

This paper contains 16 sections, 13 theorems, 75 equations, 10 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. The proposed family of MRFs
  3. Graph notations and definitions
  4. Stochastic dynamics of the proposed family
  5. Joint probability mass function and joint probability generating function
  6. Covariance
  7. Dependence ordering
  8. Sampling
  9. Sum of the components of N
  10. Distribution of the sum
  11. Stochastic ordering of the sum and impact of dependence
  12. Expected allocations
  13. Numerical example
  14. Conclusion
  15. The binomial thinning operator
  16. ...and 1 more sections

Key Result

Theorem 1

Given a tree $\mathcal{T}=(\mathcal{V},\mathcal{E})$ and a chosen root $r\in\mathcal{V}$, let $\mathcal{T}_r$ be its rooted version. Consider a vector of dependence parameters $\boldsymbol{\alpha} = (\alpha_e, e \in \mathcal{E})$, $\boldsymbol{\alpha}\in[0,1]^d$, and define $\boldsymbol{L} = (L_v, \ Then, the vector $\boldsymbol{N}$ is a tree-structured MRF whose joint distribution has Poisson mar

Figures (10)

  • Figure 1: Depiction of three typical tree shapes: (a) $d$-vertex star; (b) $d$-vertex series tree; (c) $\chi$-nary tree of radius 2.
  • Figure 2: The 7-vertex rooted tree $\mathcal{T}_1$ of Example \ref{['ex:StoDynamics']}.
  • Figure 3: Trees $\mathcal{T}$ and $\mathcal{T}^{\prime}$ of Example \ref{['ex:TreesPruning']}.
  • Figure 4: Trees $\mathcal{T}$ and $\mathcal{T}^{\prime}$ of Example \ref{['ex:NoSupermodularNShape']}.
  • Figure 5: Tree $\mathcal{T}$ for the numerical example.
  • ...and 5 more figures

Theorems & Definitions (36)

  • Definition 1: MRF
  • Theorem 1: Stochastic representation
  • proof
  • Example 1
  • Theorem 2: Choice of the root
  • proof
  • Corollary 1
  • proof
  • Example 2
  • Theorem 3: Joint probability mass function
  • ...and 26 more