Pivotal CLTs for Pseudolikelihood via Conditional Centering in Dependent Random Fields
Nabarun Deb
TL;DR
This work develops a general limit theory for conditionally centered statistics in dependent random fields, showing that under verifiable smoothness of conditional means, the centered statistic converges to a Gaussian scale mixture with a random scale driven by a quadratic-variance term and interaction effects. A data-driven studentization yields a pivotal Gaussian limit, enabling robust, non-sparse MPLE-based inference for models with dense and irregular dependencies. The framework yields new CLTs for MPLEs in Ising models (including tensor interactions) and ERGMs, including joint CLTs for inverse temperature and magnetization in dense regimes and beyond sub-criticality. The approach unifies analysis across dense graphs and irregular graphon limits, providing practical tools for inference on modern network models while using a novel method of moments with combinatorial decision-tree pruning. Overall, the results offer a versatile, exact asymptotic regime for pseudolikelihood-based inference in complex dependent structures with broad applicability in statistical physics and network modeling.
Abstract
In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}\sum_{i=1}^N c_i(g(σ_i)-\mathbb{E}_N[g(σ_i)|σ_j,j\neq i])$$ 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.