Statistical Inference for Pairwise Graphical Models Using Score Matching
Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. As a result, there is a large body of literature focused on consistent model selection. However, scientists are often interested in understanding uncertainty associated with the estimated parameters, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for edge parameters for pairwise graphical models based on Hyvärinen scoring rule. Hyvärinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form. We prove that the estimator is p n-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.
This paper studies the problem of estimating parameters in a pairwise graphical model and constructing confidence intervals for the parameters. As part of this an asymptotically normal estimator is constructed. The key progress made in this paper is that inference (i.e., confidence intervals) is done in a setting where computation of basic quantities (e.g. partition function) is intractable. (In contrast, for Gaussian graphical model one can efficiently compute the normalizing constant.)  Specifically, an estimator based on Hyvarinen score is given for estimation of a single edge in the pairwise undirected graphical model. The new scoring rule uses the conditional density of two variables given the rest. A first step forms a preliminary Markov blanket for a pair of variables, and the estimator then re-optimizes over parameter value, which has a sort of decoupling effect. The estimator is shown to satisfy asymptotic normality subject to regularity assumptions on the model (Corollary 2). Finally, experiments on simulated data show the estimator behaves as one would expect, and an experiment on protein signaling data agrees with results from previous papers analyzing that dataset.    The estimator is  
This paper was a pleasure to read: the results address a problem of significant interest, the writing is excellent, and the overall level of scholarship is high with context and background carefully placed within the broader literature. The proofs are fairly simple, making good use of existing technology developed in the cited work, but the estimator is novel.   It would be nice to remark if possible on what does the assumption SE (line 176) really mean as far as modeling power of the model.