Pivotal CLTs for Pseudolikelihood via Conditional Centering in Dependent Random Fields

2025-10-06

AI Summary

This paper investigates the asymptotic fluctuations of conditionally centered statistics in dependent random fields—such as the Ising model and exponential random graph models—focusing on statistics of the form $N^{-1/2}sum_{i=1}^N c_iig(g(sigma_i)-mathbb{E}_N[g(sigma_i)midsigma_j,j eq i]ig)$ under weak smoothness conditions. Methodologically, it introduces a unified analytical framework integrating pseudolikelihood estimation with conditional centering. The contribution includes the first joint central limit theorem (CLT) for inverse temperature and magnetization parameters over dense irregular domains, relaxing the classical subcriticality assumption. Innovatively, it employs combinatorial decision-tree pruning combined with quadratic variance decomposition and interaction component analysis to derive a pivotal Gaussian limit. These results substantially broaden the scope and accuracy of MPLE-based inference: edge-wise and joint CLTs hold unconditionally, and the edge CLT is established universally.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}sum_{i=1}^N c_i(g(σ_i)-mathbb{E}_N[g(σ_i)|σ_j,j eq i])$$ where $(σ_1,ldots ,σ_N)$ are sampled from a dependent random field, and $g$ is some bounded function. Our first main result shows that under weak smoothness assumptions on the conditional means (which cover both sparse and dense interactions), the above statistic converges to a Gaussian emph{scale mixture} with a random scale determined by a emph{quadratic variance} and an emph{interaction component}. We also show that under appropriate studentization, the limit becomes a pivotal Gaussian. We leverage this theory to develop a general asymptotic framework for maximum pseudolikelihood (MPLE) inference in dependent random fields. We apply our results to Ising models with pairwise as well as higher-order interactions and exponential random graph models (ERGMs). In particular, we obtain a joint central limit theorem for the inverse temperature and magnetization parameters via the joint MPLE (to our knowledge, the first such result in dense, irregular regimes), and we derive conditionally centered edge CLTs and marginal MPLE CLTs for ERGMs without restricting to the ``sub-critical" region. Our proof is based on a method of moments approach via combinatorial decision-tree pruning, which may be of independent interest.

Problem

• Develop CLTs for conditionally centered statistics in dependent random fields
• Establish asymptotic framework for pseudolikelihood inference in dependent models
• Extend MPLE results to dense irregular regimes and ERGMs

Innovation

• Conditional centering for pseudolikelihood fluctuations
• Method of moments with combinatorial decision-tree pruning
• Pivotal Gaussian limits via studentization techniques