# Simultaneous Inference for Pairwise Graphical Models with Generalized Score Matching

### Abstract

Probabilistic graphical models provide a flexible yet parsimonious framework for modeling dependencies among nodes in networks. There is a vast literature on parameter estimation and consistent model selection for graphical models. However, in many of the applications, scientists are also interested in quantifying the uncertainty associated with the estimated parameters and selected models, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for statistical inference on edge parameters in pairwise graphical models based on generalized Hyvarinen scoring rule. Hyvarinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form, which is a common problem for graphical models, including Ising models and truncated Gaussian graphical models. Our estimator allows us to perform statistical inference for general graphical models whereas the existing works mostly focus on statistical inference for Gaussian graphical models where finding normalizing constant is computationally tractable. Under mild conditions that are typically assumed in the literature for consistent estimation, we prove that our proposed estimator is $sqrtn$-consistent and asymptotically normal, which allows us to construct confidence intervals and build hypothesis tests for edge parameters. Moreover, we show how our proposed method can be applied to test hypotheses that involve a large number of model parameters simultaneously. We illustrate validity of our estimator through extensive simulation studies on a diverse collection of data-generating processes.

Type
Publication
Journal of Machine Learning Research
##### Ming Yu
###### PhD (2016-2020)

Ming received his PhD in Econometrics and Statistics at University of Chicago, Booth School of Business in March 2020. His research interests include high dimensional statistical inference, non-convex optimization, and reinforcement learning, with a focus on developing novel methodologies with both practical applications and theoretical guarantees.

##### Varun Gupta
###### Associate Professor of Operations Management

Varun Gupta is an Associate Professor of Operations Management.