Estimating heterogeneous graphical models for discrete data with an application to roll call voting

Abstract

We consider the problem of jointly estimating a collection of graphical models for discrete data, corresponding to several categories that share some common structure. An example for such a setting is voting records of legislators on different issues, such as defense, energy, and healthcare. We develop a Markov graphical model to characterize the heterogeneous dependence structures arising from such data. The model is fitted via a joint estimation method that preserves the underlying common graph structure, but also allows for differences between the networks. The method employs a group penalty that targets the common zero interaction effects across all the networks. We apply the method to describe the internal networks of the U.S. Senate on several important issues. Our analysis reveals individual structure for each issue, distinct from the underlying well-known bipartisan structure common to all categories which we are able to extract separately. We also establish consistency of the proposed method both for parameter estimation and model selection, and evaluate its numerical performance on a number of simulated examples.

Published at http://dx.doi.org/10.1214/13-AOAS700 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (9)

• Sparse permutation invariant covariance estimation
• High-dimensional Ising model selection using ${\ell }_1$-regularized logistic regression
• Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator
• Model Selection Through Sparse Maximum Likelihood Estimation
• Horseshoes in multidimensional scaling and local kernel methods
• High-dimensional covariance estimation by minimizing ${\ell }_1$-penalized log-determinant divergence
• Nonconcave penalized composite conditional likelihood estimation of sparse Ising models
• On the Distribution of Penalized Maximum Likelihood Estimators: The LASSO, SCAD, and Thresholding
• Improved Estimation of High-dimensional Ising Models

Cited by in corpus (3)

• Bayesian Analysis of High-dimensional Discrete Graphical Models
• Global sensitivity analysis in probabilistic graphical models
• Estimation of Graphical Models through Structured Norm Minimization