Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-Dimensional Covariate-Dependent Discrete Graphical Models and Dynamic Ising Models

Lyndsay Roach, Qiong Li, Nanwei Wang, Xin Gao

TL;DR

This work develops covariate-dependent discrete graphical models to capture time- and covariate-driven changes in dependence among high-dimensional discrete variables, with the dynamic Ising model as a special case. It combines a covariate-modulated log-linear parametrization (baseline and slope graphs) with a likelihood framework, leveraging pseudo-likelihood for scalable high-dimensional parameter estimation and scalable birth-death MCMC (SBDMCM) for structure learning. Theoretical results establish asymptotic normality of estimators, and extensive simulations validate estimation, hypothesis testing, and model selection performance. The authors demonstrate utility on a longitudinal gene-expression dataset related to influenza vaccination, illustrating covariate-driven network changes and offering a principled toolkit for dynamic, discrete networks in genomics and beyond.

Abstract

We propose a covariate-dependent discrete graphical model for capturing dynamic networks among discrete random variables, allowing the dependence structure among vertices to vary with covariates. This discrete dynamic network encompasses the dynamic Ising model as a special case. We formulate a likelihood-based approach for parameter estimation and statistical inference. We achieve efficient parameter estimation in high-dimensional settings through the use of the pseudo-likelihood method. To perform model selection, a birth-and-death Markov chain Monte Carlo algorithm is proposed to explore the model space and select the most suitable model.

High-Dimensional Covariate-Dependent Discrete Graphical Models and Dynamic Ising Models

TL;DR

This work develops covariate-dependent discrete graphical models to capture time- and covariate-driven changes in dependence among high-dimensional discrete variables, with the dynamic Ising model as a special case. It combines a covariate-modulated log-linear parametrization (baseline and slope graphs) with a likelihood framework, leveraging pseudo-likelihood for scalable high-dimensional parameter estimation and scalable birth-death MCMC (SBDMCM) for structure learning. Theoretical results establish asymptotic normality of estimators, and extensive simulations validate estimation, hypothesis testing, and model selection performance. The authors demonstrate utility on a longitudinal gene-expression dataset related to influenza vaccination, illustrating covariate-driven network changes and offering a principled toolkit for dynamic, discrete networks in genomics and beyond.

Abstract

We propose a covariate-dependent discrete graphical model for capturing dynamic networks among discrete random variables, allowing the dependence structure among vertices to vary with covariates. This discrete dynamic network encompasses the dynamic Ising model as a special case. We formulate a likelihood-based approach for parameter estimation and statistical inference. We achieve efficient parameter estimation in high-dimensional settings through the use of the pseudo-likelihood method. To perform model selection, a birth-and-death Markov chain Monte Carlo algorithm is proposed to explore the model space and select the most suitable model.

Paper Structure

This paper contains 24 sections, 2 theorems, 50 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model Setup and Statistical Inference
  3. Static discrete graphical model
  4. Covariate-dependent discrete graphical model
  5. Estimation procedure
  6. Theoretical properties
  7. Pseudo-likelihood Method for High-dimensional Covariate-dependent Discrete Graphical Models
  8. Methodology
  9. High-dimensional dynamic Ising model
  10. Parameter estimation for high-dimensional dynamic Ising model
  11. Bayesian Model Selection for Covariate-Dependent Discrete Graphical Models
  12. Birth-death MCMC method for dynamic Ising model
  13. Birth-death rate computation
  14. Scalable birth-death MCMC(SBDMCM) for high-dimensional dynamic Ising models
  15. Simulations
  16. ...and 9 more sections

Key Result

Theorem 1

(Asymptotic Normality). Under Assumptions 1, there exists a local maximizer $\hat{\boldsymbol{\theta}}$ of $\ell_n(\boldsymbol{\theta})$ such that $\left\|\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}^*\right\|_2=O_p\left(n^{-\frac{1}{2}}\right)$. Furthermore, the maximum likelihood estimator $\hat{

Figures (8)

  • Figure 1: Graphs (a) and (b) are visualizations of $G(2)$ and $G(4)$, respectively.
  • Figure 2: Graphs (a) and (b) are visualizations of $G(2)_{0}$ and $G(2)_{1}$, respectively.
  • Figure 3: Graphs (a) and (b) are visualizations of $G(4)_{0}$ and $G(4)_{1}$, respectively.
  • Figure 4: Dynamic Ising model structures for the simulation study
  • Figure 5: Relative MSE v.s. sample size for the model in Figure \ref{['fig:isingstruc']}
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2