Estimating Densities with Non-Parametric Exponential Families
This addresses the issue of model degeneracy in exponential random graph models for small-size graph distributions, though it appears incremental as it builds on existing exponential family and kernel density estimation methods.
The paper tackles the problem of density estimation when the true density may not lie within a chosen exponential family by augmenting sufficient statistics with features to accumulate probability mass near observed points, resulting in a non-parametric model that approximates densities outside the family and modifies the exponential random graph model to address model degeneracy.
We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-parametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential family. Using the proposed approach, we modify the exponential random graph model, commonly used for modeling small-size graph distributions, to address the well-known issue of model degeneracy.