Computing Pseudolikelihood Estimators for Exponential-Family Random Graph Models

Abstract

The reputation of the maximum pseudolikelihood estimator (MPLE) for Exponential Random Graph Models (ERGM) has undergone a drastic change over the past 30 years. While first receiving broad support, mainly due to its computational feasibility and the lack of alternatives, general opinions started to change with the introduction of approximate maximum likelihood estimator (MLE) methods that became practicable due to increasing computing power and the introduction of MCMC methods. Previous comparison studies appear to yield contradicting results regarding the preference of these two point estimators; however, there is consensus that the prevailing method to obtain an MPLE’s standard error by the inverse Hessian matrix generally underestimates standard errors. We propose replacing the inverse Hessian matrix by an approximation of the Godambe matrix that results in confidence intervals with appropriate coverage rates and that, in addition, enables examining for model degeneracy. Our results also provide empirical evidence for the asymptotic normality of the MPLE under certain conditions.

Editorial: Symposium Data Science and Statistics 2022

“…They present an empirical evaluation of this testing method for linear regression, and briefly discuss an extension of this method to nonlinear applications. Schmid and Hunter (2023) proposes to estimate maximum pseudolikelihood estimator standard errors for exponential random graph models using an estimated Godambe matrix. Their results provide empirical evidence for the asymptotic normality of the maximum pseudolikelihood estimator under certain conditions.…”

Approximate Bayesian computation using asymptotically normal point estimates

“…, K, with these MPLEs adjusted by network size by specifying in each ERGM an offset term |x| (edge count of network x) with fixed coefficient log 1 n (Krivitsky et al 2011). The MPLE (e.g., Schmid & Hunter, 2023) was introduced for lattice models (Besag, 1974), and developed to estimate ERGM parameters (Strauss & Ikeda, 1990;Frank & Strauss, 1986) because the exact MLE of the ERGM is intractable for a binary network with more than a handful of nodes n, as mentioned above. Specifically, for example, if the observed network dataset x is an undirected binary network, x = (x i,j ∈ {0, 1}) n×n on n nodes, described by scalar or vector valued statistics g k (x) for k = 1, .…”